Calculus

Problem 16301

Calculate the integral: $\int_{0}^{\frac{\pi}{4}} \cos ^{4} 2 x \, dx$ .

See Solution

Problem 16302

You want a 30-year mortgage of \ $275,000. Interpret$ P(2)=1016.45 $and$ P'(2)=137.52$. What do they mean?

See Solution

Problem 16303

Calculate the integral of the function: $\int\left(-6 x^{3}+9 x^{2}+4 x-3\right) d x$ .

See Solution

Problem 16304

Find the integral of the function: $\int(5 x^{2}-8 x+5) \, dx$

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Problem 16305

Calculate the work done on a gas mixture expanding from 91.0 L to 96.0 L at 39.0 atm. Answer in $\square \mathrm{kJ}$ .

See Solution

Problem 16306

Find the integral of the function: $\int\left(x^{\frac{3}{2}}+2 x+3\right) d x$ .

See Solution

Problem 16307

Evaluate the integral: $\int\left(\sqrt{x}+\frac{1}{3 \sqrt{x}}\right) d x$

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Problem 16308

Find the derivative of $f(x)=3 \sqrt{x}+\frac{2}{\sqrt{x}}-5$ .

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Problem 16309

Calculate the integral $\int\left(y^{2} \sqrt[3]{y} \, dy\right)$ .

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Problem 16310

How fast is the distance from a car on a highway to a farmhouse increasing when the car is 2 miles past the intersection? The farmhouse is 8 miles away and the car travels at $75 \mathrm{mph}$ .

See Solution

Problem 16311

A 17-foot ladder leans against a wall. If the top slips down at 3 ft/s, how fast is the foot moving when the top is 15 ft up? The foot moves at $\mathrm{ft/s}$ .

See Solution

Problem 16312

Calculez la masse de Polonium-210 après 365 jours avec $m(t)=300 e^{-\frac{\ln 2}{140} t}$ et trouvez le temps pour atteindre 200 $\mathrm{mg}$ .

See Solution

Problem 16313

A balloon's volume increases at $3.7 \mathrm{ft}^{3} / \mathrm{min}$ . Find the diameter's rate of increase at 1.6 ft.

See Solution

Problem 16314

Find the rate of change of demand $x$ when price $p = 10$ and $dp/dt = 2$ using $2x^2 - 12xp + 50p^2 = 8200$ .

See Solution

Problem 16315

Calculate the integral for the area under the curve from 0 to 5: $\int_{0}^{5}(5-x) d x=$

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Problem 16316

A jogger runs on a circular track with radius $50 \mathrm{ft}$ . At $(30,40)$ , her $x$ -coordinate changes at $20 \mathrm{ft/s}$ . Find $d y / d t$ .

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Problem 16317

Find the average value $f_{\text{ave}}$ of $f(x)=\sqrt{16-x^{2}}$ from $a=0$ to $b=4$ , rounded to 3 decimal places.

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Problem 16318

Given the function $r(x)=\frac{3 x^{2}+1}{(x-4)^{2}}$ , complete the tables for values of $x$ . Describe the behavior near $x=4$ and find the horizontal asymptote.

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Problem 16319

You’re taking a \ $250,000 mortgage. Interpret$ P(6)=1498.88 $and$ P'(6)=160.73$. What do they mean?

See Solution

Problem 16320

Déterminez le max/min de $f(t)=10 \sin (4 \pi t)$ et les temps où $f(t)$ atteint ces valeurs entre 0 et 3 s.

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Problem 16321

Find the volume generated by revolving the curve $y=3 x^{2}$ around the $y$ -axis, rounded to one decimal place.

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Problem 16322

Find the tangent line equation and $\frac{d^{2} y}{d x^{2}}$ for $x=\sec t$ , $y=\cos t$ at $t=\frac{\pi}{4}$ .

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Problem 16323

Find the tangent line equation at $t=-\frac{\pi}{4}$ for $x=\sin t, y=\cos t$ . Also, calculate $\frac{d^{2} y}{d x^{2}}$ .
Tangent line: $y=\square x+\square$ .

See Solution

Problem 16324

Find the tangent line equation at $t=\frac{\pi}{2}$ for the curve defined by $x=-\cos t$ , $y=4+\sin t$ , and $\frac{d^{2} y}{d x^{2}}$ at that point.

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Problem 16325

Find the tangent line equation at $t=11$ for $x=\frac{1}{t+10}$ and $y=\frac{t}{t-10}$ . Also, calculate $\frac{d^{2} y}{d x^{2}}$ .
Write the tangent line as: $y=\square x-\square$

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Problem 16326

Find the tangent line equation at $t=-1$ for $x=2t^{3}+9$ , $y=t^{6}$ and $\frac{d^{2} y}{d x^{2}}$ at that point.
Write the equation of the tangent line $y=\square$

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Problem 16327

Find the work to pump water from a cylindrical tank (radius $5 \mathrm{ft}$ , height $20 \mathrm{ft}$ ) filled to $5 \mathrm{ft}$ .

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Problem 16328

Find the surface area from revolving $x=t+2\sqrt{3}, y=\frac{t^{2}}{2}+2\sqrt{3}t+3$ about the $y$ -axis.

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Problem 16329

Find the work to pump water from a cylindrical tank (radius $5 \mathrm{ft}$ , height $20 \mathrm{ft}$ ) out the top. Show integration steps.

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Problem 16330

Find the tangent line equation at $t=10$ for the curve defined by $x=\frac{1}{t+9}, y=\frac{t}{t-9}$ . Also, find $\frac{d^{2} y}{d x^{2}}$ at this point.
Write the tangent line as: $y=\square x-\square$

See Solution

Problem 16331

Find the work to pump water from a cylindrical tank (radius $5 \mathrm{ft}$ , height $20 \mathrm{ft}$ , water height $5 \mathrm{ft}$ ) out. Round to whole ft-lb.

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Problem 16332

Given $f\left(\frac{\pi}{4}\right)=-6$ and $f^{\prime}\left(\frac{\pi}{4}\right)=5$ , find $g^{\prime}(\pi / 4)$ and $h^{\prime}(\pi / 4)$ for $g(x)=f(x) \sin x$ and $h(x)=\frac{\cos x}{f(x)}$ .

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Problem 16333

Find $f^{\prime}(1)$ for $f(x)=h(g(x) k(x))$ with given values: $g(1)=7$ , $k(1)=0$ , $h(1)=-8$ , $g^{\prime}(1)=1$ , $k^{\prime}(1)=-7$ , $h^{\prime}(1)=2$ , $h^{\prime}(0)=5$ .

See Solution

Problem 16334

Find $\left(f^{-1}\right)^{\prime}(1)$ for the one-to-one function $f(x)=2x+\cos(x)$ using the formula $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}(f^{-1}(x))}$ .

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Problem 16335

Find the work done in stretching a spring from $10 \mathrm{~cm}$ to $18 \mathrm{~cm}$ with a $12 \mathrm{~N}$ force.

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Problem 16336

Find the area between the curve $y=6 \cos x$ and the $x$ -axis from $x=\pi / 6.2$ to $x=\pi / 2$ using integration. Round to one decimal place.

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Problem 16337

Find the area between the curve $y=\left(8 / x^{2}\right)$ , lines $x=1$ , $x=7.5$ , and the axis. Round to one decimal place.

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Problem 16338

Find the area between the curve $y=6 \cos (x)$ and the $x$ -axis from $x=\frac{\pi}{6}$ to $x=2\pi$ .

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Problem 16339

A balloon deflates at $12 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ . Find the surface area change rate when volume is $36 \pi \mathrm{cm}^{3}$ . Use $V=\frac{4}{3} \pi r^{3}$ and $A=4 \pi r^{2}$ .

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Problem 16340

Find the sum to infinity of the series $\frac{1}{5} + \frac{1}{25} + \frac{1}{125} + \frac{1}{625} + \ldots$

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Problem 16341

Find $c$ using the Mean Value Theorem for Integrals for $f(x)=\cos x$ over $\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$ .

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Problem 16342

Find the value(s) of $c$ from the Mean Value Theorem for Integrals for $f(x)=\cos x$ on $\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$ .

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Problem 16343

Find the displacement and total distance for the velocity function $v(t)=t^{3}-6 t^{2}+11 t-6$ for $1 \leq t \leq 4$ .

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Problem 16344

Evaluate the limit $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(c_{i}\right) \Delta x_{i}$ for $f(x)=\sqrt{x}$ , $y=0$ , $x=0$ , $x=15$ . Round to three decimal places. Hint: $c_{i}=\frac{15 i^{2}}{n^{2}}$ .

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Problem 16345

Berechne die Integrale und finde den Wert für $Z$ in den folgenden Aufgaben: a) $\int_{0}^{Z} x d x=18$ , b) $\int_{1}^{Z} 4 x d x=30$ , c) $\int_{Z}^{10} 2 x d x=19$ , d) $\int_{0}^{2 z} 0,4 d x=8$ .

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Problem 16346

Berechne den Wert von $z$ aus den Integralen: a) $\int_{0}^{2} x \, dx=18$ , b) $\int_{1}^{2} 4x \, dx=30$ , c) $\int_{z}^{10} 2x \, dx=19$ , d) $\int_{0}^{2z} 0,4 \, dx=8$ .

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Problem 16347

Berechnen Sie die Grenze $Z$ für die Integrale: a) $\int_{0}^{Z} x d x=18$ , b) $\int_{1}^{Z} 4 x d x=30$ , c) $\int_{Z}^{10} 2 x d x=19$ , d) $\int_{0}^{2 z} 0,4 d x=8$ .

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Problem 16348

En jeger skyter horisontalt mot en blink $180 \mathrm{~cm}$ over bakken, $9.81$ meter unna. Hvor langt under midten treffer kulen? $\mathrm{cm}$

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Problem 16349

Simplify the expression using the quotient rule: $\frac{x^{3}}{x}$ , assuming bases are not equal to 0.

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Problem 16350

Berechnen Sie die Integrale: a) $\int_{0}^{3}\left(x^{2}-2\right) d x$ , b) $\int_{-3}^{6}\left(5-x^{2}\right) d x$ , c) $\int_{0}^{6}\left(x^{3}-2 x^{2}\right) d x$ .

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Problem 16351

Zeigen Sie, dass $F$ eine Stammfunktion von $f$ ist und nennen Sie drei weitere Stammfunktionen für die gegebenen $f$ .

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Problem 16352

Zeigen Sie, dass $F$ eine Stammfunktion von $f$ ist und nennen Sie drei weitere Stammfunktionen für die gegebenen $f$ .

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Problem 16353

Zeigen Sie, dass $\mathrm{F}$ eine Stammfunktion von $\mathrm{f}$ ist und nennen Sie drei weitere Stammfunktionen von $f$ .

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Problem 16354

Berechnen Sie die Integrale: d) $\int_{-2}^{0}\left(-2 x^{3}+3 x^{2}-4\right) d x$ e) $\int_{-1}^{1}\left(x^{5}-5 x^{4}\right) d x$ f) $\int_{0}^{10}\left(\frac{x^{2}-3 x}{5}+1\right) d x$

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Problem 16355

Calculate the integral from 2 to 3 of the function $1+\frac{1}{x^{2}}$ .

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Problem 16356

Überprüfen Sie, ob die Integrale $\int_{-5}^{10} F_{A}'(x) \, dx$ und $\int_{-5}^{10} F_{K}'(x) \, dx$ gleich sind.

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Problem 16357

Berechnen Sie die Integrale: g) $\int_{1}^{2}\left(0,5 x^{4}-5 x\right) dx$ , h) $\int_{-2}^{0}\left(\frac{1}{3} x^{3}-\frac{1}{2} x^{2}\right) dx$ , i) $\int_{0}^{2 \pi} \cos (x) dx$ .

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Problem 16358

Find the second derivative of $f(x)$ if $f^{\prime}(x)=-12(2-3x)^{3}$ .

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Problem 16359

Find the dimensions of a cylinder with maximum volume inscribed in a cone of height 4 and base radius 7. Radius = $R$ , Height = $H$ .

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Problem 16360

Bearbeite die Funktionen $f(x)=x^{3}$ und $f(x)=x^{4}$ , finde den Differenzenquotienten und die Ableitungsfunktion.

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Problem 16361

Berechne die Flächeninhalte unter der Funktion $g(x)=0,2 x+1$ für die Intervalle $[0 ; 3]$ , $[2 ; 5]$ und finde eine Stammfunktion.

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Problem 16362

Find the value of $k$ such that $\int_{1}^{2} k x^{3} d x=75$ .

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Problem 16363

How long (in years) will it take for \$6000 to grow to \$40000 at a 6% annual interest rate, compounded continuously? Round to the nearest tenth.

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Problem 16364

Sei $k$ so gewählt, dass die Flächeninhalte $A_{1}$ und $A_{2}$ zwischen $f(x)=\frac{1}{4} x^{3}$ und der $x$ -Achse im Verhältnis 1:15 sind.

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Problem 16365

Berechnen Sie die Umlaufdauer $T$ eines Satelliten in $500 \mathrm{~km}$ Höhe. Nutzen Sie den Mondabstand von $384000 \mathrm{~km}$ und seine Umlaufzeit von 27,3 Tagen.

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Problem 16366

Bestimme die Ableitung der Funktionen: a) $f(x)=e^{7x}$ , b) $f(x)=e^{2x}$ , c) $f(x)=e^{-2x}$ , d) $f(x)=e^{0.2x}$ , e) $f(x)=3e^{2x}$ , f) $f(x)=0.5e^{4x}$ .

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Problem 16367

Bestimmen Sie die Ableitung der Funktionen und klammern Sie aus: a) $f(x)=x \cdot e^{x}$ b) $f(x)=(x-3) \cdot e^{x}$

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Problem 16368

Gegeben ist die Funktion $f_{a}(x)=\frac{1}{8} x^{4}-\frac{a}{12} x^{3}+2 x$ . Untersuchen Sie das Verhalten und die Extrempunkte von $f_{0}$ .

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Problem 16369

Find the horizontal asymptotes for the rational function $f(x)=\frac{-x^{2}+2 x-2}{x^{2}-12 x+27}$ .

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Problem 16370

Find the average rate of change of $f(x)=\frac{16}{(x-4)^{2}}+5$ from $x=1$ to $x=2$ .

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Problem 16371

Calculate the indefinite integral and include the constant of integration $C$ :
$\int (x-7)^{2} \, dx$

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Problem 16372

A 2.00 kg rock is dropped from a 30.0 m building. Find its momentum when it hits the ground.

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Problem 16373

Gegeben ist die Funktion $f_{a}(x)=\frac{1}{8} x^{4}-\frac{a}{12} x^{3}+2 x$ . Untersuchen Sie das Verhalten von $f_{0}$ und dessen Ableitungen.

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Problem 16374

Find the derivative of $f(x)=x^{2} \tan^{-1} x$ .

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Problem 16375

Evaluate the integral from -1 to 2 of $(x^{2}-3 x)$ . Show all steps.

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Problem 16376

Find the derivative of the function $f(x)=(\ln (x))^{\sec (x)}$ . What is $f^{\prime}(x)$ ?

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Problem 16377

Berechnen Sie das unbestimmte Integral für die Funktionen: a) $f(x)=\frac{2 x^{2}+3}{x^{2}}$ , b) $f(x)=12 x^{3}+4 x^{2}$ , d) $f(x)=\frac{2 x^{4}-5}{x^{2}}$ , e) $f(x)=2 x^{3}-\frac{4}{\sqrt{x}}$ .

See Solution

Problem 16378

Find the derivative $\frac{d y}{d x}$ for $y=3 \frac{1}{\sin ^{4}(\ln (x))}$ . What is $\frac{d y}{d x}=\square$ ?

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Problem 16379

Bestimmen Sie die unbestimmten Integrale der folgenden Funktionen: a) $f(x)=10 x^{4}$ , b) $f(x)=x^{2}-6 x+4$ , c) $f(x)=x+\sqrt{x}$ , d) $f(x)=\frac{4}{x^{2}}$ , e) $f(x)=\frac{2 x^{2}+3}{5^{2}}$ , f) $f(x)=12 x^{3}+4 x^{2}$ , g) $f(x)=\frac{2 x^{4}-5}{x^{2}}$ , h) $f(x)=2 x^{3}-\frac{4}{\sqrt{x}}$ .

See Solution

Problem 16380

Berechnen Sie die Integrale: a) $\int_{1}^{2}(x^{2}+1) d x$ b) $\int_{-1}^{2}(x-2) d x$ c) $\int_{0}^{3}(2-\frac{1}{2} x^{2}) d x$

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Problem 16381

Berechnen Sie die Integrale: a) $\int_{1}^{2}\left(x^{2}+1\right) dx$ , b) $\int_{-1}^{2}(x-2) dx$ , c) $\int_{0}^{3}\left(2-\frac{1}{2} x^{2}\right) dx$ .

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Problem 16382

Find the derivative of $f(x)=2x-\sqrt{4x+5}$ using the limit definition. Then, find the tangent and normal line at $x=1$ .

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Problem 16383

Berechne mit CAS die Schnittpunkte von $f$ und $g$ mit den Achsen, Extrem- und Wendepunkte sowie die Tangente bei $x=2$ für
$f(x)=2 \cdot x^{4}+7 x^{3}+5 x^{2}, \quad g(x)=e^{x} \cdot(x-2).$

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Problem 16384

Find the derivative of $f(x)=2x-\sqrt{4x+5}$ using limits, then find tangent and normal line equations at $x=1$ .

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Problem 16385

Determine the half-life of a radioactive element with decay function $A(t)=A_{0} e^{-0.0376 t}$ .

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Problem 16386

Find the half-life of a radioactive substance with a decay rate of $k = -0.029$ per year. Round to one decimal place.

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Problem 16387

Find the 2nd and 3rd derivatives of $f(x) = e^{\sin x + \tan x}$ and evaluate at $x=1$ .

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Problem 16388

Finde die 3 Ableitungen von $f(x)=x^{3}-6 x^{2}+9 x-2$ .

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Problem 16389

Leiten Sie die Funktionen ab und klammern Sie aus: a) $f(x)=2 x \cdot e^{3 x}$ , b) $g(x)=(x-3) \cdot e^{2 x}$ , c) $f(t)=t^{2} \cdot e^{\frac{1}{2} t}+5$ , d) $g(t)=(t^{3}-2 t^{2}) \cdot e^{-2 t}$ , e) $f(x)=(2 x-4) \cdot e^{0,5 x}$ , f) $h(x)=(x^{2}-x) \cdot e^{-0,01 x}$ .

See Solution

Problem 16390

Leiten Sie die Funktionen ab und vereinfachen Sie, wenn möglich: a) $f(x)=2 x+e^{2 x}$ b) $f(x)=e^{2 x}-5 \cdot e^{\frac{1}{5} x}$ c) $f(x)=x^{2}+e^{3 x-1}$ d) $f(x)=20 \cdot e^{0,1 x}+x+5$ e) $f(x)=\frac{1}{3} x^{3}-2 \cdot e^{-0,25 x}$ f) $f(x)=2 x^{2}-e^{-x+3}+x^{3}$

See Solution

Problem 16391

Bestimmen Sie die Ableitung und klammern Sie aus für: a) $f(x)=x e^{x}$ , b) $f(x)=(x-3)e^{x}$ , e) $f(x)=\sin(x)e^{x}$ , f) $f(x)=(x^{2}-5)e^{x}$ , c) $f(x)=x^{2}e^{x}$ , d) $f(x)=x e^{x}-x^{2}$ , g) $f(x)=x^{4}e^{x}-7$ , h) $f(x)=5x-x e^{x}$ .

See Solution

Problem 16392

Finde die Ableitungen von $f(x)=x^{3}-6 x^{2}+9 x-2$ .

See Solution

Problem 16393

Gegeben sind die Funktionen $f(x)=2x^{4}+7x^{3}+5x^{2}$ und $g(x)=e^{x}(x-2)$ .
1. Bestimmen Sie mit einem CAS die Schnittpunkte mit den Achsen, Extrem- und Wendepunkte.
2. Finden Sie die Tangentengleichung an $x=2$ .

See Solution

Problem 16394

Find the derivative of $y=\sqrt[3]{x+\sqrt{6 x+7}}+\sec(5x)-(2x+4x^2)^{\frac{1}{4}}$ .

See Solution

Problem 16395

Find the derivative of $y=\left(\frac{x}{x+2}\right)^{2}\left(\frac{x^{2}+3}{4+2 x}\right)$ .

See Solution

Problem 16396

Test if the series $\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{4^{n}}$ converges or diverges, and find the sum if it converges.

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Problem 16397

Find the derivative of $y=\left(\frac{u}{u-1}\right)^{2}-\left(\frac{u}{u+1}\right)^{2}$ .

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Problem 16398

Untersuchen Sie die Ableitung von $f(x)=\ln (x)$ und überprüfen Sie, ob $f^{\prime}(x)=\frac{1}{x}$ für $x>0$ gilt.

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Problem 16399

Find the derivative of $y=\ln \left(\frac{x^{3}-x^{2}-x+3}{x^{4}+x^{2}+x+1}\right)$ .

See Solution

Problem 16400

Berechnen Sie das Integral $\int \frac{1}{x} d x$ für die Funktion $f(x)=\frac{1}{x}$ .

See Solution

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Calculus

Problem 16301

Calculate the integral: ∫0π4cos⁡42x dx\int_{0}^{\frac{\pi}{4}} \cos ^{4} 2 x \, dx∫04π​​cos42xdx.

Problem 16302

You want a 30-year mortgage of \275,000.Interpret275,000. Interpret 275,000.InterpretP(2)=1016.45and and andP'(2)=137.52$. What do they mean?

Problem 16303

Calculate the integral of the function: ∫(−6x3+9x2+4x−3)dx\int\left(-6 x^{3}+9 x^{2}+4 x-3\right) d x∫(−6x3+9x2+4x−3)dx.

Problem 16304

Find the integral of the function: ∫(5x2−8x+5) dx\int(5 x^{2}-8 x+5) \, dx∫(5x2−8x+5)dx

Problem 16305

Calculate the work done on a gas mixture expanding from 91.0 L to 96.0 L at 39.0 atm. Answer in □kJ\square \mathrm{kJ}□kJ.

Problem 16306

Find the integral of the function: ∫(x32+2x+3)dx\int\left(x^{\frac{3}{2}}+2 x+3\right) d x∫(x23​+2x+3)dx.

Problem 16307

Evaluate the integral: ∫(x+13x)dx\int\left(\sqrt{x}+\frac{1}{3 \sqrt{x}}\right) d x∫(x​+3x​1​)dx

Problem 16308

Find the derivative of f(x)=3x+2x−5f(x)=3 \sqrt{x}+\frac{2}{\sqrt{x}}-5f(x)=3x​+x​2​−5.

Problem 16309

Calculate the integral ∫(y2y3 dy)\int\left(y^{2} \sqrt[3]{y} \, dy\right)∫(y23y​dy).

Problem 16310

How fast is the distance from a car on a highway to a farmhouse increasing when the car is 2 miles past the intersection? The farmhouse is 8 miles away and the car travels at 75mph75 \mathrm{mph}75mph.

Problem 16311

A 17-foot ladder leans against a wall. If the top slips down at 3 ft/s, how fast is the foot moving when the top is 15 ft up? The foot moves at ft/s\mathrm{ft/s}ft/s.

Problem 16312

Calculez la masse de Polonium-210 après 365 jours avec m(t)=300e−ln⁡2140tm(t)=300 e^{-\frac{\ln 2}{140} t}m(t)=300e−140ln2​t et trouvez le temps pour atteindre 200 mg\mathrm{mg}mg.

Problem 16313

A balloon's volume increases at 3.7ft3/min3.7 \mathrm{ft}^{3} / \mathrm{min}3.7ft3/min. Find the diameter's rate of increase at 1.6 ft.

Problem 16314

Find the rate of change of demand xxx when price p=10p = 10p=10 and dp/dt=2dp/dt = 2dp/dt=2 using 2x2−12xp+50p2=82002x^2 - 12xp + 50p^2 = 82002x2−12xp+50p2=8200.

Problem 16315

Calculate the integral for the area under the curve from 0 to 5: ∫05(5−x)dx=\int_{0}^{5}(5-x) d x=∫05​(5−x)dx=

Problem 16316

A jogger runs on a circular track with radius 50ft50 \mathrm{ft}50ft. At (30,40)(30,40)(30,40), her xxx-coordinate changes at 20ft/s20 \mathrm{ft/s}20ft/s. Find dy/dtd y / d tdy/dt.

Problem 16317

Find the average value favef_{\text{ave}}fave​ of f(x)=16−x2f(x)=\sqrt{16-x^{2}}f(x)=16−x2​ from a=0a=0a=0 to b=4b=4b=4, rounded to 3 decimal places.

Problem 16318

Given the function r(x)=3x2+1(x−4)2r(x)=\frac{3 x^{2}+1}{(x-4)^{2}}r(x)=(x−4)23x2+1​, complete the tables for values of xxx. Describe the behavior near x=4x=4x=4 and find the horizontal asymptote.

Problem 16319

You’re taking a \250,000mortgage.Interpret250,000 mortgage. Interpret 250,000mortgage.InterpretP(6)=1498.88and and andP'(6)=160.73$. What do they mean?

Problem 16320

Déterminez le max/min de f(t)=10sin⁡(4πt)f(t)=10 \sin (4 \pi t)f(t)=10sin(4πt) et les temps où f(t)f(t)f(t) atteint ces valeurs entre 0 et 3 s.

Problem 16321

Find the volume generated by revolving the curve y=3x2y=3 x^{2}y=3x2 around the yyy-axis, rounded to one decimal place.

Problem 16322

Find the tangent line equation and d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for x=sec⁡tx=\sec tx=sect, y=cos⁡ty=\cos ty=cost at t=π4t=\frac{\pi}{4}t=4π​.

Problem 16323

Find the tangent line equation at t=−π4t=-\frac{\pi}{4}t=−4π​ for x=sin⁡t,y=cos⁡tx=\sin t, y=\cos tx=sint,y=cost. Also, calculate d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​. Tangent line: y=□x+□y=\square x+\squarey=□x+□.

Problem 16324

Find the tangent line equation at t=π2t=\frac{\pi}{2}t=2π​ for the curve defined by x=−cos⁡tx=-\cos tx=−cost, y=4+sin⁡ty=4+\sin ty=4+sint, and d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ at that point.

Problem 16325

Find the tangent line equation at t=11t=11t=11 for x=1t+10x=\frac{1}{t+10}x=t+101​ and y=tt−10y=\frac{t}{t-10}y=t−10t​. Also, calculate d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​. Write the tangent line as: y=□x−□ y=\square x-\square y=□x−□

Problem 16326

Find the tangent line equation at t=−1t=-1t=−1 for x=2t3+9x=2t^{3}+9x=2t3+9, y=t6y=t^{6}y=t6 and d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ at that point. Write the equation of the tangent line y=□ y=\square y=□

Problem 16327

Find the work to pump water from a cylindrical tank (radius 5ft5 \mathrm{ft}5ft, height 20ft20 \mathrm{ft}20ft) filled to 5ft5 \mathrm{ft}5ft.

Problem 16328

Find the surface area from revolving x=t+23,y=t22+23t+3x=t+2\sqrt{3}, y=\frac{t^{2}}{2}+2\sqrt{3}t+3x=t+23​,y=2t2​+23​t+3 about the yyy-axis.

Problem 16329

Find the work to pump water from a cylindrical tank (radius 5ft5 \mathrm{ft}5ft, height 20ft20 \mathrm{ft}20ft) out the top. Show integration steps.

Problem 16330

Find the tangent line equation at t=10t=10t=10 for the curve defined by x=1t+9,y=tt−9x=\frac{1}{t+9}, y=\frac{t}{t-9}x=t+91​,y=t−9t​. Also, find d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ at this point. Write the tangent line as: y=□x−□ y=\square x-\square y=□x−□

Problem 16331

Find the work to pump water from a cylindrical tank (radius 5ft5 \mathrm{ft}5ft, height 20ft20 \mathrm{ft}20ft, water height 5ft5 \mathrm{ft}5ft) out. Round to whole ft-lb.

Problem 16332

Problem 16333

Problem 16334

Find (f−1)′(1)\left(f^{-1}\right)^{\prime}(1)(f−1)′(1) for the one-to-one function f(x)=2x+cos⁡(x)f(x)=2x+\cos(x)f(x)=2x+cos(x) using the formula (f−1)′(x)=1f′(f−1(x))\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}(f^{-1}(x))}(f−1)′(x)=f′(f−1(x))1​.

Problem 16335

Find the work done in stretching a spring from 10 cm10 \mathrm{~cm}10 cm to 18 cm18 \mathrm{~cm}18 cm with a 12 N12 \mathrm{~N}12 N force.

Problem 16336

Find the area between the curve y=6cos⁡xy=6 \cos xy=6cosx and the xxx-axis from x=π/6.2x=\pi / 6.2x=π/6.2 to x=π/2x=\pi / 2x=π/2 using integration. Round to one decimal place.

Problem 16337

Find the area between the curve y=(8/x2)y=\left(8 / x^{2}\right)y=(8/x2), lines x=1x=1x=1, x=7.5x=7.5x=7.5, and the axis. Round to one decimal place.

Problem 16338

Find the area between the curve y=6cos⁡(x)y=6 \cos (x)y=6cos(x) and the xxx-axis from x=π6x=\frac{\pi}{6}x=6π​ to x=2πx=2\pix=2π.

Problem 16339

A balloon deflates at 12 cm3 s−112 \mathrm{~cm}^{3} \mathrm{~s}^{-1}12 cm3 s−1. Find the surface area change rate when volume is 36πcm336 \pi \mathrm{cm}^{3}36πcm3. Use V=43πr3V=\frac{4}{3} \pi r^{3}V=34​πr3 and A=4πr2A=4 \pi r^{2}A=4πr2.

Problem 16340

Find the sum to infinity of the series 15+125+1125+1625+… \frac{1}{5} + \frac{1}{25} + \frac{1}{125} + \frac{1}{625} + \ldots 51​+251​+1251​+6251​+…

Problem 16341

Calculate the integral: $\int_{0}^{\frac{\pi}{4}} \cos ^{4} 2 x \, dx$ .

You want a 30-year mortgage of \ $275,000. Interpret$ P(2)=1016.45 $and$ P'(2)=137.52$. What do they mean?

Calculate the integral of the function: $\int\left(-6 x^{3}+9 x^{2}+4 x-3\right) d x$ .

Find the integral of the function: $\int(5 x^{2}-8 x+5) \, dx$

Calculate the work done on a gas mixture expanding from 91.0 L to 96.0 L at 39.0 atm. Answer in $\square \mathrm{kJ}$ .

Find the integral of the function: $\int\left(x^{\frac{3}{2}}+2 x+3\right) d x$ .

Evaluate the integral: $\int\left(\sqrt{x}+\frac{1}{3 \sqrt{x}}\right) d x$

Find the derivative of $f(x)=3 \sqrt{x}+\frac{2}{\sqrt{x}}-5$ .

Calculate the integral $\int\left(y^{2} \sqrt[3]{y} \, dy\right)$ .

How fast is the distance from a car on a highway to a farmhouse increasing when the car is 2 miles past the intersection? The farmhouse is 8 miles away and the car travels at $75 \mathrm{mph}$ .

A 17-foot ladder leans against a wall. If the top slips down at 3 ft/s, how fast is the foot moving when the top is 15 ft up? The foot moves at $\mathrm{ft/s}$ .

Calculez la masse de Polonium-210 après 365 jours avec $m(t)=300 e^{-\frac{\ln 2}{140} t}$ et trouvez le temps pour atteindre 200 $\mathrm{mg}$ .

A balloon's volume increases at $3.7 \mathrm{ft}^{3} / \mathrm{min}$ . Find the diameter's rate of increase at 1.6 ft.

Find the rate of change of demand $x$ when price $p = 10$ and $dp/dt = 2$ using $2x^2 - 12xp + 50p^2 = 8200$ .

Calculate the integral for the area under the curve from 0 to 5: $\int_{0}^{5}(5-x) d x=$

A jogger runs on a circular track with radius $50 \mathrm{ft}$ . At $(30,40)$ , her $x$ -coordinate changes at $20 \mathrm{ft/s}$ . Find $d y / d t$ .

Find the average value $f_{\text{ave}}$ of $f(x)=\sqrt{16-x^{2}}$ from $a=0$ to $b=4$ , rounded to 3 decimal places.

Given the function $r(x)=\frac{3 x^{2}+1}{(x-4)^{2}}$ , complete the tables for values of $x$ . Describe the behavior near $x=4$ and find the horizontal asymptote.

You’re taking a \ $250,000 mortgage. Interpret$ P(6)=1498.88 $and$ P'(6)=160.73$. What do they mean?

Déterminez le max/min de $f(t)=10 \sin (4 \pi t)$ et les temps où $f(t)$ atteint ces valeurs entre 0 et 3 s.

Find the volume generated by revolving the curve $y=3 x^{2}$ around the $y$ -axis, rounded to one decimal place.

Find the tangent line equation and $\frac{d^{2} y}{d x^{2}}$ for $x=\sec t$ , $y=\cos t$ at $t=\frac{\pi}{4}$ .

Find the tangent line equation at $t=-\frac{\pi}{4}$ for $x=\sin t, y=\cos t$ . Also, calculate $\frac{d^{2} y}{d x^{2}}$ .
Tangent line: $y=\square x+\square$ .

Find the tangent line equation at $t=\frac{\pi}{2}$ for the curve defined by $x=-\cos t$ , $y=4+\sin t$ , and $\frac{d^{2} y}{d x^{2}}$ at that point.

Find the tangent line equation at $t=11$ for $x=\frac{1}{t+10}$ and $y=\frac{t}{t-10}$ . Also, calculate $\frac{d^{2} y}{d x^{2}}$ .
Write the tangent line as: $y=\square x-\square$

Find the tangent line equation at $t=-1$ for $x=2t^{3}+9$ , $y=t^{6}$ and $\frac{d^{2} y}{d x^{2}}$ at that point.
Write the equation of the tangent line $y=\square$

Find the work to pump water from a cylindrical tank (radius $5 \mathrm{ft}$ , height $20 \mathrm{ft}$ ) filled to $5 \mathrm{ft}$ .

Find the surface area from revolving $x=t+2\sqrt{3}, y=\frac{t^{2}}{2}+2\sqrt{3}t+3$ about the $y$ -axis.

Find the work to pump water from a cylindrical tank (radius $5 \mathrm{ft}$ , height $20 \mathrm{ft}$ ) out the top. Show integration steps.

Find the tangent line equation at $t=10$ for the curve defined by $x=\frac{1}{t+9}, y=\frac{t}{t-9}$ . Also, find $\frac{d^{2} y}{d x^{2}}$ at this point.
Write the tangent line as: $y=\square x-\square$

Find the work to pump water from a cylindrical tank (radius $5 \mathrm{ft}$ , height $20 \mathrm{ft}$ , water height $5 \mathrm{ft}$ ) out. Round to whole ft-lb.

Find $\left(f^{-1}\right)^{\prime}(1)$ for the one-to-one function $f(x)=2x+\cos(x)$ using the formula $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}(f^{-1}(x))}$ .

Find the work done in stretching a spring from $10 \mathrm{~cm}$ to $18 \mathrm{~cm}$ with a $12 \mathrm{~N}$ force.

Find the area between the curve $y=6 \cos x$ and the $x$ -axis from $x=\pi / 6.2$ to $x=\pi / 2$ using integration. Round to one decimal place.

Find the area between the curve $y=\left(8 / x^{2}\right)$ , lines $x=1$ , $x=7.5$ , and the axis. Round to one decimal place.

Find the area between the curve $y=6 \cos (x)$ and the $x$ -axis from $x=\frac{\pi}{6}$ to $x=2\pi$ .

A balloon deflates at $12 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ . Find the surface area change rate when volume is $36 \pi \mathrm{cm}^{3}$ . Use $V=\frac{4}{3} \pi r^{3}$ and $A=4 \pi r^{2}$ .

Find the sum to infinity of the series $\frac{1}{5} + \frac{1}{25} + \frac{1}{125} + \frac{1}{625} + \ldots$

Find $c$ using the Mean Value Theorem for Integrals for $f(x)=\cos x$ over $\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$ .

Find the value(s) of $c$ from the Mean Value Theorem for Integrals for $f(x)=\cos x$ on $\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$ .

Find the displacement and total distance for the velocity function $v(t)=t^{3}-6 t^{2}+11 t-6$ for $1 \leq t \leq 4$ .

Berechne die Integrale und finde den Wert für $Z$ in den folgenden Aufgaben: a) $\int_{0}^{Z} x d x=18$ , b) $\int_{1}^{Z} 4 x d x=30$ , c) $\int_{Z}^{10} 2 x d x=19$ , d) $\int_{0}^{2 z} 0,4 d x=8$ .

Berechne den Wert von $z$ aus den Integralen: a) $\int_{0}^{2} x \, dx=18$ , b) $\int_{1}^{2} 4x \, dx=30$ , c) $\int_{z}^{10} 2x \, dx=19$ , d) $\int_{0}^{2z} 0,4 \, dx=8$ .

Berechnen Sie die Grenze $Z$ für die Integrale: a) $\int_{0}^{Z} x d x=18$ , b) $\int_{1}^{Z} 4 x d x=30$ , c) $\int_{Z}^{10} 2 x d x=19$ , d) $\int_{0}^{2 z} 0,4 d x=8$ .

En jeger skyter horisontalt mot en blink $180 \mathrm{~cm}$ over bakken, $9.81$ meter unna. Hvor langt under midten treffer kulen? $\mathrm{cm}$

Simplify the expression using the quotient rule: $\frac{x^{3}}{x}$ , assuming bases are not equal to 0.

Berechnen Sie die Integrale: a) $\int_{0}^{3}\left(x^{2}-2\right) d x$ , b) $\int_{-3}^{6}\left(5-x^{2}\right) d x$ , c) $\int_{0}^{6}\left(x^{3}-2 x^{2}\right) d x$ .

Zeigen Sie, dass $F$ eine Stammfunktion von $f$ ist und nennen Sie drei weitere Stammfunktionen für die gegebenen $f$ .

Zeigen Sie, dass $F$ eine Stammfunktion von $f$ ist und nennen Sie drei weitere Stammfunktionen für die gegebenen $f$ .

Zeigen Sie, dass $\mathrm{F}$ eine Stammfunktion von $\mathrm{f}$ ist und nennen Sie drei weitere Stammfunktionen von $f$ .

Calculate the integral from 2 to 3 of the function $1+\frac{1}{x^{2}}$ .

Überprüfen Sie, ob die Integrale $\int_{-5}^{10} F_{A}'(x) \, dx$ und $\int_{-5}^{10} F_{K}'(x) \, dx$ gleich sind.

Find the second derivative of $f(x)$ if $f^{\prime}(x)=-12(2-3x)^{3}$ .

Find the dimensions of a cylinder with maximum volume inscribed in a cone of height 4 and base radius 7. Radius = $R$ , Height = $H$ .

Bearbeite die Funktionen $f(x)=x^{3}$ und $f(x)=x^{4}$ , finde den Differenzenquotienten und die Ableitungsfunktion.

Berechne die Flächeninhalte unter der Funktion $g(x)=0,2 x+1$ für die Intervalle $[0 ; 3]$ , $[2 ; 5]$ und finde eine Stammfunktion.

Find the value of $k$ such that $\int_{1}^{2} k x^{3} d x=75$ .

Sei $k$ so gewählt, dass die Flächeninhalte $A_{1}$ und $A_{2}$ zwischen $f(x)=\frac{1}{4} x^{3}$ und der $x$ -Achse im Verhältnis 1:15 sind.

Berechnen Sie die Umlaufdauer $T$ eines Satelliten in $500 \mathrm{~km}$ Höhe. Nutzen Sie den Mondabstand von $384000 \mathrm{~km}$ und seine Umlaufzeit von 27,3 Tagen.

Bestimme die Ableitung der Funktionen: a) $f(x)=e^{7x}$ , b) $f(x)=e^{2x}$ , c) $f(x)=e^{-2x}$ , d) $f(x)=e^{0.2x}$ , e) $f(x)=3e^{2x}$ , f) $f(x)=0.5e^{4x}$ .

Bestimmen Sie die Ableitung der Funktionen und klammern Sie aus: a) $f(x)=x \cdot e^{x}$ b) $f(x)=(x-3) \cdot e^{x}$

Gegeben ist die Funktion $f_{a}(x)=\frac{1}{8} x^{4}-\frac{a}{12} x^{3}+2 x$ . Untersuchen Sie das Verhalten und die Extrempunkte von $f_{0}$ .

Find the horizontal asymptotes for the rational function $f(x)=\frac{-x^{2}+2 x-2}{x^{2}-12 x+27}$ .

Find the average rate of change of $f(x)=\frac{16}{(x-4)^{2}}+5$ from $x=1$ to $x=2$ .

Calculate the indefinite integral and include the constant of integration $C$ :
$\int (x-7)^{2} \, dx$

Gegeben ist die Funktion $f_{a}(x)=\frac{1}{8} x^{4}-\frac{a}{12} x^{3}+2 x$ . Untersuchen Sie das Verhalten von $f_{0}$ und dessen Ableitungen.

Find the derivative of $f(x)=x^{2} \tan^{-1} x$ .