Calculus

Problem 32301

Find $y^{\prime}(e)$ if $y=x^{\ln x}$ .

See Solution

Problem 32302

Find the derivative of $f(x)=\ln(2 e^{-x} \sin x)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32303

Find the derivative $\frac{d y}{d x}$ for $y=\sin \left(e^{x^{2}}\right)$ .

See Solution

Problem 32304

Find the derivative $f^{\prime}(0)$ for $f(x)=\ln (\sec (x)+\tan (x))$ . Choose from: 1, -1, 0, $\frac{\pi}{2}$ .

See Solution

Problem 32305

Find the derivative $f^{\prime}(0)$ for the function $f(x)=\ln (\sec (x)+\tan (x))$ . Choices: $\frac{\pi}{2}$ , 1, 0, $-1$ .

See Solution

Problem 32306

Find the integral $\int \csc (5 x) d x$ and choose the correct answer from the options given.

See Solution

Problem 32307

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ given $x^{2}+y^{2}=1$ . Choices include: $\frac{-x}{y^{2}}$ , $\frac{x^{2}-y^{2}}{y^{3}}$ , $\frac{x^{2}+y^{2}}{y^{3}}$ , $\frac{-1}{y^{3}}$ .

See Solution

Problem 32308

Evaluate the integral of $\frac{\pi}{6} \sec (2 x) d x$ . Choose the correct answer from the options provided.

See Solution

Problem 32309

Find $f^{\prime}(2)$ for the function $f(x)=\ln \left(x^{2}-3\right)$ . Choices: $e$ , 4, 1, 2.

See Solution

Problem 32310

Find $\frac{d y}{d x}$ at the point (1,1) for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

See Solution

Problem 32311

Find $\frac{d y}{d x}$ for the equation $e^{y}=6 x+3 y$ . Choose the correct option from the list.

See Solution

Problem 32312

Find the tangent line equation to $f(x)=e^{2 x}$ at the point $(0,1)$ . Choose from the options.

See Solution

Problem 32313

2

See Solution

Problem 32314

Find $f^{\prime}(0)$ for $f(x)=\int_{1}^{e^{2 x}} t^{2}+\ln t \, dt$ . Write a number only.

See Solution

Problem 32315

Find $f^{\prime}(e)$ for the function $f(x)=x^{\ln x}$ . Options: 2, 0, 1, $e$ .

See Solution

Problem 32316

Given $x^{2}+y^{2}=1$ , find the second derivative $\frac{d^{2} y}{d x^{2}}$ . Choices are:
1. $\frac{-1}{y^{3}}$
2. $\frac{-x}{y^{2}}$
3. $\frac{x^{2}-y^{2}}{y^{3}}$
4. $\frac{x^{2}+y^{2}}{y^{3}}$

See Solution

Problem 32317

Find the slope of the tangent line for $f(x)=e^{\tan x}$ at $x=0$ .

See Solution

Problem 32318

Find the derivative of the function $f(x)=\ln \left(e^{-x}+1\right)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32319

Find the integral of $\sec(5x) \, dx$ and choose the correct answer from the options provided.

See Solution

Problem 32320

Find the derivative of $f(x)=\ln(2 e^{-x} \sin x)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32321

Find the integral of $\csc(5x) \, dx$ .

See Solution

Problem 32322

Calculate the integral $\int_{e}^{e^{2}} \frac{4(\ln x)^{3}}{x} dx$ . Provide just the numerical answer.

See Solution

Problem 32323

Find the first derivative $\frac{d y}{d x}$ at the point $(1,1)$ for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ . Options: 2, 0, $\frac{1}{2}$ , -2.

See Solution

Problem 32324

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ . Choices: $-1$ , 1, $\frac{-1}{e}$ , e.

See Solution

Problem 32325

Find the derivative of $y=\sin(e^{x^{2}})$ with respect to $x$ : $\frac{dy}{dx}$ .

See Solution

Problem 32326

Find the first derivative $\frac{d y}{d x}$ at the point (1,1) for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

See Solution

Problem 32327

Find the integral of $\sec(5x) \, dx$ . Choose the correct answer from the options provided.

See Solution

Problem 32328

Find $\frac{d y}{d x}$ for the equation $e^{y}=6 x+3 y$ .

See Solution

Problem 32329

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ if $x^{2}+y^{2}=1$ . Options: A) $\frac{-1}{y^{3}}$ B) $\frac{x^{2}+y^{2}}{y^{3}}$ C) $\frac{x^{2}-y^{2}}{y^{3}}$ D) $\frac{-x}{y^{2}}$

See Solution

Problem 32330

Find the derivative $f^{\prime}(x)$ if $f(x)=\ln(2 e^{-x} \sin x)$ .

See Solution

Problem 32331

Find $\frac{d y}{d x}$ for $x^{2} y^{4}-2 y=3$ at the point $(2,1)$ . Options: 2, -2, 20, $\frac{1}{2}$ .

See Solution

Problem 32332

Find $\frac{d y}{d x}$ for $x=y \cos (2 x)$ at the point $\left(\frac{\pi}{2}, \frac{-\pi}{2}\right)$ . Options: $-1$ , $\frac{1}{2}$ , 0, 1.

See Solution

Problem 32333

Find $f^{\prime}(e)$ if $f(x)=x^{\ln x}$ . Choices: 1, 0, e, 2.

See Solution

Problem 32334

Evaluate the integral $\int_{0}^{1} x^{2} 2^{x^{3}} d x$ and choose between: a. $\frac{1}{\ln (4)}$ b. $\frac{1}{\ln (8)}$

See Solution

Problem 32335

Find the integral $\int \csc (5 x) d x$ and choose the correct answer from the options provided.

See Solution

Problem 32336

Find the derivative of $f(x)=\ln(2 e^{-x} \sin x)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32337

Find $\left(f^{-1}\right)^{\prime}(-3)$ given $f(-3)=5, f(4)=-3, f^{\prime}(-3)=\frac{-1}{4}, f^{\prime}(4)=\frac{1}{2}$ .

See Solution

Problem 32338

Find $\frac{d y}{d x}$ if $e^{y}=6 x+3 y$ . Options: $\frac{6 x}{e^{y}+3}$ , $\frac{6}{e^{y}+3}$ , $\frac{6}{e^{y}+3 y}$ , $\frac{6}{e^{y}-3}$ .

See Solution

Problem 32339

Find $\frac{d y}{d x}$ for $y=\ln \sqrt{\tan x}$ at $x=\frac{\pi}{4}$ . What is the value?

See Solution

Problem 32340

2

See Solution

Problem 32341

Find $\frac{d y}{d x}$ for the equation $e^{y}=6 x+3 y$ . Options: $\frac{6}{e^{y}+3}$ , $\frac{6 x}{e^{y+3}}$ , $\frac{6}{e^{y}-3}$ , $\frac{6}{e^{y}+3 y}$ .

See Solution

Problem 32342

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

See Solution

Problem 32343

Find the tangent line to the curve $f(x)=e^{2 x}$ at the point $(0,1)$ . Choose the correct equation.

See Solution

Problem 32344

Find $\frac{d y}{d x}$ for $y=\ln \sqrt{\tan x}$ at $x=\frac{\pi}{4}$ .

See Solution

Problem 32345

Find $\frac{d y}{d x}$ at the point (1,1) for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

See Solution

Problem 32346

Find the derivative of $f(x)=\ln(e^{-x}+1)$ : $f^{\prime}(x)=?$ Choose from: $\frac{-1}{1+e^{x}}$ , $\frac{e^{-x}}{e^{-x}+1}$ , $\frac{1}{e^{-x}+1}$ , $\frac{-1}{e^{-x}+1}$ .

See Solution

Problem 32347

Calculate the integral $\int_{0}^{\frac{\pi}{6}} \sec (2 x) d x$ and choose the correct answer from the options.

See Solution

Problem 32348

Solve the differential equation: $y^{\prime \prime}+3 y^{\prime}-40 y=6 x^{2}$ .

See Solution

Problem 32349

Bestimmen Sie die Definitionsmenge der Funktionen und die mittlere Änderungsrate in den Intervallen $I_{1}=[-1 ; 0], I_{2}=[0 ; 1], I_{3}=[1 ; 3], I_{4}=[0 ; 3]$ für $f$ .

See Solution

Problem 32350

Calculate the integral $\int_{0}^{1} 9 x^{2} 2^{x^{3}} d x$ . Choose from: (a) $\frac{6}{\ln 2}$ , (b) $\frac{1}{\ln 2}$ , (c) $\ln 2$ , (d) $\frac{3}{\ln 2}$ .

See Solution

Problem 32351

Find the derivative of $x e^{x}-e^{x}$ and choose the correct option.

See Solution

Problem 32352

إذا كانت $\Delta ص = \frac{8هـ - هـ^2}{4}$ ، فما قيمة $ق'(هـ)$ في الفترة $[5, 5+هـ]$ ؟

See Solution

Problem 32353

A conical container (radius $9$ ft, height $36$ ft) has liquid at $31$ ft. Find work to pump it to rim and $5$ ft above rim.

See Solution

Problem 32354

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\frac{8}{9+h}-\frac{8}{9}}{h}$ by direct substitution. Simplify to $\lim _{h \rightarrow 0}(\square)$ .

See Solution

Problem 32355

Find the radius of a loop for a pilot in an aerobatic plane experiencing weightlessness at $310 \mathrm{~m/s}$ with gravity $9.8 \mathrm{~m/s}^2$ . Answer in $\mathrm{km}$ .

See Solution

Problem 32356

Wheat is pumped into a silo at $f(t)=8+5 \cos \left(\frac{t}{25}\right)$ and removed at $g(t)=\sqrt{t+1}$ .
a. Gallons pumped in 60 min? b. Find $f^{\prime}(60)$ and explain its meaning. c. Is wheat amount increasing or decreasing at $t=60$ ? Why? d. Write $A(t)$ for total gallons in the silo. e. Find wheat amount after 120 min using $A(t)$ . f. When is wheat amount maximum in $[0, 120]$ ? Justify.

See Solution

Problem 32357

Solve the equation $\frac{d R}{d x}=a\left(R^{2}+1\right)$ for $R$ , using constant $C$ .

See Solution

Problem 32358

Calculate $W=\frac{62 \cdot 4 \pi}{48} \cdot \int_{0}^{31} (36 h^{2}-h^{3}) \, dh$ .

See Solution

Problem 32359

Find the derivative of $f(x)=\ln(2 e^{-x} \sin x)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32360

33. Eingesperrtes Rechteck: Bestimme die Koordinaten von Punkt P auf $f(x)=1+\frac{4}{x^{2}}$ , um a) Fläche und b) Umfang minimal zu halten.

See Solution

Problem 32361

Bestimmen Sie die Stammfunktion $F(x) = (a \cdot x + b) e^{-0,5x}$ von $f$ durch Koeffizientenvergleich und berechnen Sie $A$ über $I=[0, 7]$ .

See Solution

Problem 32362

Find the derivative of $y=e^{x}(\sin x+\cos x)$ with respect to $x$ .

See Solution

Problem 32363

Find the tangent line to the curve $f(x)=e^{2 x}$ at the point $(0,1)$ . Select the correct equation.

See Solution

Problem 32364

Find $f^{\prime}(e)$ if $f(x)=x^{\ln x}$ . Choices: 0, 1, $e$ , 2.

See Solution

Problem 32365

Find the tangent line to $f(x) = e^{-5x}$ at $x = 0$ and the area of the triangle it forms in the first quadrant.

See Solution

Problem 32366

Solve the differential equation $7x - 8y\sqrt{x^2 + 1} \frac{dy}{dx} = 0$ with $y(0) = 1$ . Find $y =$ .

See Solution

Problem 32367

-5

See Solution

Problem 32368

Find $f'(0)$ if $f(x)=\int_{1}^{e^{2 x}} (t^{2}+\ln t) dt$ . Write a number only Answer:

See Solution

Problem 32369

Bestimme die unbestimmten Integrale: a) $\int 7 \, dx$ b) $\int\left(\frac{1}{3} x^{2}+5 x-3\right) \, dx$

See Solution

Problem 32370

Evaluate the integral $\int \sqrt{x^{2}-25} d x$ and choose the best technique: A, B, C, D, or E?

See Solution

Problem 32371

Find the first derivative $\frac{d y}{d x}$ of $\sin y = x^{3}+y^{2}+8$ at the point $(-2,0)$ .

See Solution

Problem 32372

Find the relative maximum point of $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x+1$ . Use $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ .

See Solution

Problem 32373

Berechnen Sie die Integrale: a) $\int_{0}^{2}(2 x+1) d x$ , b) $\int_{-1}^{3} 0,5 x^{3} d x$ , c) $\int_{0}^{4}\left(x^{2}-\frac{x}{3}\right) d x$ .

See Solution

Problem 32374

Evaluate the limit: $\lim _{x \rightarrow 9} \frac{\sqrt{x}-3}{x-9}$ by simplifying it for direct substitution.

See Solution

Problem 32375

Find the relative maximum point of the function $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x+1$ . Use $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ .

See Solution

Problem 32376

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{225+h}-15}{h}$ using direct substitution. Simplify your answer.

See Solution

Problem 32377

Evaluate the integral $\int \cos ^{2}(x) d x$ . Which method is most efficient? A. $u=\sin(x)$ B. $\cos ^{2}(x)=\frac{1}{2}(1+\cos(2x))$ C. $u=\cos(x)$ D. Not integrable E. $\frac{1}{3}\cos^{3}(x)+C$

See Solution

Problem 32378

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{8 x}{5 x-7}$ . If yes, find its value.

See Solution

Problem 32379

Find the derivative of the inverse function at 1, given $f(-1)=1$ , $f(1)=0$ , $f'(-1)=\frac{1}{5}$ , $f'(1)=\frac{-1}{5}$ . Write a number only.
Answer: $-5$

See Solution

Problem 32380

Determine if the limit exists:
$\lim _{x \rightarrow \infty} \frac{3 x^{3}+2 x-2}{5 x^{4}-9 x^{3}-7}$
Choose A (find value) or B (limit does not exist).

See Solution

Problem 32381

Is it true or false that if $f(x)$ is continuous at $x=a$ , then $\lim_{x \rightarrow a} f(x)=f(a)$ ? Explain.

See Solution

Problem 32382

Differentiate these expressions with respect to $x$ : $y=(4 x^{3}+5 x)^{2}$ and $y=(6 x^{3}-5 x)^{-2}$ .

See Solution

Problem 32383

Find $f^{\prime}(2)$ using the limit definition of the derivative for $f(x)=x^{2}+5x$ .

See Solution

Problem 32384

Find the limit as $x$ approaches -2 for the expression $\frac{\sqrt{x^{2}+5}-3}{x+2}$ .

See Solution

Problem 32385

Find the limit: $\lim _{x \rightarrow 0} \frac{(x+1)^{2}-1}{x}$ for $x \neq 0$ .

See Solution

Problem 32386

Find the volume of a solid between $y=0$ and $y=2$ with circular cross-sections from $x=\sqrt{10} y^{2}$ .

See Solution

Problem 32387

Find the volume of a solid between planes at $y=0$ and $y=2$ with circular cross-sections defined by $x=\sqrt{10} y^{2}$ .

See Solution

Problem 32388

Find the 3rd iteration interval for $h(x) = x^4 - 3x + 1$ using the bisection method on $[0,1]$ .

See Solution

Problem 32389

Find the mass $M$ and center of mass $\bar{x}$ of a wire from $-1$ to $1$ with density $\delta(x)=7+3 x^{2}$ . Set up the mass integral.

See Solution

Problem 32390

Determinați constanta $a$ pentru ca $f(x)$ să fie densitate de probabilitate și scrieți funcția de repartiție.

See Solution

Problem 32391

Max untersucht die Ableitung von $f(x)=x^{6}$ als Produkt $x^{2} \cdot x^{4}$ . Vervollständigen Sie die Tabelle und formulieren Sie eine Ableitungsregel.

See Solution

Problem 32392

Finde die $x$ -Werte, wo die Ableitungen der Funktionen $g(x)$ , $h(x)$ , $p(x)$ , $q(x)$ und $r(x)$ gleich 0 sind.

See Solution

Problem 32393

Bestimmen Sie die Extremstellen der Funktion $f(x) = (3x - 9) \cdot e^{0.5x}$ .

See Solution

Problem 32394

Max untersucht die Ableitung von $f(x)=x^{6}$ als Produkt $u(x) \cdot v(x)$ .
a) Vervollständige die Tabelle für Ableitungen.
b) Formuliere die Ableitungsregel für $f(x) = u(x) \cdot v(x)$ .

See Solution

Problem 32395

Ergänzen Sie die Ableitungen für die Funktionen: a) $f(x)=(3x-1)e^{x}$ , b) $g(x)=5x^{2}e^{x}$ , c) $f(x)=(1-3x)(2-x)$ . Geben Sie zuerst $u(x)$ und $v(x)$ an und leiten Sie dann ab.

See Solution

Problem 32396

Vervollständigen Sie die Ableitungen für die Funktionen $f(x)$ und $g(x)$ sowie die Produkte $u(x)$ und $v(x)$ .

See Solution

Problem 32397

Leiten Sie die Funktionen ab, nutzen Sie die Produktregel:
1. $f(x)=(3 x-1) \cdot e^{x}, f^{\prime}(x)=3 e^{x}+$
2. $g(x)=5 x^{2} \cdot e^{x}, g^{\prime}(x)=10 x+$
3. $f(x)=(1-3 x) \cdot(2-x)=-3(2-x)+$
Geben Sie $u(x)$ und $v(x)$ an und leiten Sie ab:
a) $f(x)=\left(2 x^{3}-x\right) \cdot e^{x}$ b) $f(x)=\left(x^{2}-1\right)\left(2 x^{2}+5\right)$ c) $f(x)=\left(5 x^{2}-4\right) \cdot e^{x}$

See Solution

Problem 32398

Find the derivative $y^{\prime}$ if $y=\sqrt{\ln \sqrt{t}}$ . Choose from the options provided.

See Solution

Problem 32399

Bestimmen Sie die Koordinaten und Art des lokalen Extrempunkts der Funktion $f(x)=1,6 \cdot(5-x) \cdot e^{x-4}$ . Finde Punkt $Q$ mit maximalem Anstieg für $0 \leq x_{Q} \leq 4$ .

See Solution

Problem 32400

Determine if the series $\sum_{n=1}^{\infty} \frac{n^{3}}{n^{4}+z}$ converges or diverges for fixed $z \in \mathbb{C} \backslash \mathbb{R}$ .

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337

Calculus

Problem 32301

Find y′(e)y^{\prime}(e)y′(e) if y=xln⁡xy=x^{\ln x}y=xlnx.

Problem 32302

Find the derivative of f(x)=ln⁡(2e−xsin⁡x)f(x)=\ln(2 e^{-x} \sin x)f(x)=ln(2e−xsinx). What is f′(x)f^{\prime}(x)f′(x)?

Problem 32303

Find the derivative dydx\frac{d y}{d x}dxdy​ for y=sin⁡(ex2)y=\sin \left(e^{x^{2}}\right)y=sin(ex2).

Problem 32304

Find the derivative f′(0)f^{\prime}(0)f′(0) for f(x)=ln⁡(sec⁡(x)+tan⁡(x))f(x)=\ln (\sec (x)+\tan (x))f(x)=ln(sec(x)+tan(x)). Choose from: 1, -1, 0, π2\frac{\pi}{2}2π​.

Problem 32305

Find the derivative f′(0)f^{\prime}(0)f′(0) for the function f(x)=ln⁡(sec⁡(x)+tan⁡(x))f(x)=\ln (\sec (x)+\tan (x))f(x)=ln(sec(x)+tan(x)). Choices: π2\frac{\pi}{2}2π​, 1, 0, −1-1−1.

Problem 32306

Find the integral ∫csc⁡(5x)dx\int \csc (5 x) d x∫csc(5x)dx and choose the correct answer from the options given.

Problem 32307

Find the second derivative d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ given x2+y2=1x^{2}+y^{2}=1x2+y2=1. Choices include: −xy2\frac{-x}{y^{2}}y2−x​, x2−y2y3\frac{x^{2}-y^{2}}{y^{3}}y3x2−y2​, x2+y2y3\frac{x^{2}+y^{2}}{y^{3}}y3x2+y2​, −1y3\frac{-1}{y^{3}}y3−1​.

Problem 32308

Evaluate the integral of π6sec⁡(2x)dx\frac{\pi}{6} \sec (2 x) d x6π​sec(2x)dx. Choose the correct answer from the options provided.

Problem 32309

Find f′(2)f^{\prime}(2)f′(2) for the function f(x)=ln⁡(x2−3)f(x)=\ln \left(x^{2}-3\right)f(x)=ln(x2−3). Choices: eee, 4, 1, 2.

Problem 32310

Find dydx\frac{d y}{d x}dxdy​ at the point (1,1) for the equation x23+y23=2yx^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}x32​+y32​=2y​.

Problem 32311

Find dydx\frac{d y}{d x}dxdy​ for the equation ey=6x+3ye^{y}=6 x+3 yey=6x+3y. Choose the correct option from the list.

Problem 32312

Find the tangent line equation to f(x)=e2xf(x)=e^{2 x}f(x)=e2x at the point (0,1)(0,1)(0,1). Choose from the options.

Problem 32313

2

Problem 32314

Find f′(0)f^{\prime}(0)f′(0) for f(x)=∫1e2xt2+ln⁡t dtf(x)=\int_{1}^{e^{2 x}} t^{2}+\ln t \, dtf(x)=∫1e2x​t2+lntdt. Write a number only.

Problem 32315

Find f′(e)f^{\prime}(e)f′(e) for the function f(x)=xln⁡xf(x)=x^{\ln x}f(x)=xlnx. Options: 2, 0, 1, eee.

Problem 32316

Given x2+y2=1x^{2}+y^{2}=1x2+y2=1, find the second derivative d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​. Choices are: 1. −1y3\frac{-1}{y^{3}}y3−1​ 2. −xy2\frac{-x}{y^{2}}y2−x​ 3. x2−y2y3\frac{x^{2}-y^{2}}{y^{3}}y3x2−y2​ 4. x2+y2y3\frac{x^{2}+y^{2}}{y^{3}}y3x2+y2​

Problem 32317

Find the slope of the tangent line for f(x)=etan⁡xf(x)=e^{\tan x}f(x)=etanx at x=0x=0x=0.

Problem 32318

Find the derivative of the function f(x)=ln⁡(e−x+1)f(x)=\ln \left(e^{-x}+1\right)f(x)=ln(e−x+1). What is f′(x)f^{\prime}(x)f′(x)?

Problem 32319

Find the integral of sec⁡(5x) dx\sec(5x) \, dxsec(5x)dx and choose the correct answer from the options provided.

Problem 32320

Find the derivative of f(x)=ln⁡(2e−xsin⁡x)f(x)=\ln(2 e^{-x} \sin x)f(x)=ln(2e−xsinx). What is f′(x)f^{\prime}(x)f′(x)?

Problem 32321

Find the integral of csc⁡(5x) dx\csc(5x) \, dxcsc(5x)dx.

Problem 32322

Calculate the integral ∫ee24(ln⁡x)3xdx\int_{e}^{e^{2}} \frac{4(\ln x)^{3}}{x} dx∫ee2​x4(lnx)3​dx. Provide just the numerical answer.

Problem 32323

Find the first derivative dydx\frac{d y}{d x}dxdy​ at the point (1,1)(1,1)(1,1) for the equation x23+y23=2yx^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}x32​+y32​=2y​. Options: 2, 0, 12\frac{1}{2}21​, -2.

Problem 32324

Find the slope of the tangent line of f(x)=etan⁡xf(x)=e^{\tan x}f(x)=etanx at x=0x=0x=0. Choices: −1-1−1, 1, −1e\frac{-1}{e}e−1​, e.

Problem 32325

Find the derivative of y=sin⁡(ex2)y=\sin(e^{x^{2}})y=sin(ex2) with respect to xxx: dydx\frac{dy}{dx}dxdy​.

Problem 32326

Find the first derivative dydx\frac{d y}{d x}dxdy​ at the point (1,1) for the equation x23+y23=2yx^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}x32​+y32​=2y​.

Problem 32327

Find the integral of sec⁡(5x) dx\sec(5x) \, dxsec(5x)dx. Choose the correct answer from the options provided.

Problem 32328

Find dydx\frac{d y}{d x}dxdy​ for the equation ey=6x+3ye^{y}=6 x+3 yey=6x+3y.

Problem 32329

Find the second derivative d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ if x2+y2=1x^{2}+y^{2}=1x2+y2=1. Options: A) −1y3\frac{-1}{y^{3}}y3−1​ B) x2+y2y3\frac{x^{2}+y^{2}}{y^{3}}y3x2+y2​ C) x2−y2y3\frac{x^{2}-y^{2}}{y^{3}}y3x2−y2​ D) −xy2\frac{-x}{y^{2}}y2−x​

Problem 32330

Find the derivative f′(x)f^{\prime}(x)f′(x) if f(x)=ln⁡(2e−xsin⁡x)f(x)=\ln(2 e^{-x} \sin x)f(x)=ln(2e−xsinx).

Problem 32331

Find dydx\frac{d y}{d x}dxdy​ for x2y4−2y=3x^{2} y^{4}-2 y=3x2y4−2y=3 at the point (2,1)(2,1)(2,1). Options: 2, -2, 20, 12\frac{1}{2}21​.

Problem 32332

Find dydx\frac{d y}{d x}dxdy​ for x=ycos⁡(2x)x=y \cos (2 x)x=ycos(2x) at the point (π2,−π2)\left(\frac{\pi}{2}, \frac{-\pi}{2}\right)(2π​,2−π​). Options: −1-1−1, 12\frac{1}{2}21​, 0, 1.

Problem 32333

Find f′(e)f^{\prime}(e)f′(e) if f(x)=xln⁡xf(x)=x^{\ln x}f(x)=xlnx. Choices: 1, 0, e, 2.

Problem 32334

Evaluate the integral ∫01x22x3dx\int_{0}^{1} x^{2} 2^{x^{3}} d x∫01​x22x3dx and choose between: a. 1ln⁡(4)\frac{1}{\ln (4)}ln(4)1​ b. 1ln⁡(8)\frac{1}{\ln (8)}ln(8)1​

Problem 32335

Find the integral ∫csc⁡(5x)dx\int \csc (5 x) d x∫csc(5x)dx and choose the correct answer from the options provided.

Problem 32336

Find the derivative of f(x)=ln⁡(2e−xsin⁡x)f(x)=\ln(2 e^{-x} \sin x)f(x)=ln(2e−xsinx). What is f′(x)f^{\prime}(x)f′(x)?

Problem 32337

Find (f−1)′(−3)\left(f^{-1}\right)^{\prime}(-3)(f−1)′(−3) given f(−3)=5,f(4)=−3,f′(−3)=−14,f′(4)=12f(-3)=5, f(4)=-3, f^{\prime}(-3)=\frac{-1}{4}, f^{\prime}(4)=\frac{1}{2}f(−3)=5,f(4)=−3,f′(−3)=4−1​,f′(4)=21​.

Problem 32338

Find dydx\frac{d y}{d x}dxdy​ if ey=6x+3ye^{y}=6 x+3 yey=6x+3y. Options: 6xey+3\frac{6 x}{e^{y}+3}ey+36x​, 6ey+3\frac{6}{e^{y}+3}ey+36​, 6ey+3y\frac{6}{e^{y}+3 y}ey+3y6​, 6ey−3\frac{6}{e^{y}-3}ey−36​.

Problem 32339

Find dydx\frac{d y}{d x}dxdy​ for y=ln⁡tan⁡xy=\ln \sqrt{\tan x}y=lntanx​ at x=π4x=\frac{\pi}{4}x=4π​. What is the value?

Problem 32340

Find $y^{\prime}(e)$ if $y=x^{\ln x}$ .

Find the derivative of $f(x)=\ln(2 e^{-x} \sin x)$ . What is $f^{\prime}(x)$ ?

Find the derivative $\frac{d y}{d x}$ for $y=\sin \left(e^{x^{2}}\right)$ .

Find the derivative $f^{\prime}(0)$ for $f(x)=\ln (\sec (x)+\tan (x))$ . Choose from: 1, -1, 0, $\frac{\pi}{2}$ .

Find the derivative $f^{\prime}(0)$ for the function $f(x)=\ln (\sec (x)+\tan (x))$ . Choices: $\frac{\pi}{2}$ , 1, 0, $-1$ .

Find the integral $\int \csc (5 x) d x$ and choose the correct answer from the options given.

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ given $x^{2}+y^{2}=1$ . Choices include: $\frac{-x}{y^{2}}$ , $\frac{x^{2}-y^{2}}{y^{3}}$ , $\frac{x^{2}+y^{2}}{y^{3}}$ , $\frac{-1}{y^{3}}$ .

Evaluate the integral of $\frac{\pi}{6} \sec (2 x) d x$ . Choose the correct answer from the options provided.

Find $f^{\prime}(2)$ for the function $f(x)=\ln \left(x^{2}-3\right)$ . Choices: $e$ , 4, 1, 2.

Find $\frac{d y}{d x}$ at the point (1,1) for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

Find $\frac{d y}{d x}$ for the equation $e^{y}=6 x+3 y$ . Choose the correct option from the list.

Find the tangent line equation to $f(x)=e^{2 x}$ at the point $(0,1)$ . Choose from the options.

Find $f^{\prime}(0)$ for $f(x)=\int_{1}^{e^{2 x}} t^{2}+\ln t \, dt$ . Write a number only.

Find $f^{\prime}(e)$ for the function $f(x)=x^{\ln x}$ . Options: 2, 0, 1, $e$ .

Given $x^{2}+y^{2}=1$ , find the second derivative $\frac{d^{2} y}{d x^{2}}$ . Choices are:
1. $\frac{-1}{y^{3}}$
2. $\frac{-x}{y^{2}}$
3. $\frac{x^{2}-y^{2}}{y^{3}}$
4. $\frac{x^{2}+y^{2}}{y^{3}}$

Find the slope of the tangent line for $f(x)=e^{\tan x}$ at $x=0$ .

Find the derivative of the function $f(x)=\ln \left(e^{-x}+1\right)$ . What is $f^{\prime}(x)$ ?

Find the integral of $\sec(5x) \, dx$ and choose the correct answer from the options provided.

Find the derivative of $f(x)=\ln(2 e^{-x} \sin x)$ . What is $f^{\prime}(x)$ ?

Find the integral of $\csc(5x) \, dx$ .

Calculate the integral $\int_{e}^{e^{2}} \frac{4(\ln x)^{3}}{x} dx$ . Provide just the numerical answer.

Find the first derivative $\frac{d y}{d x}$ at the point $(1,1)$ for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ . Options: 2, 0, $\frac{1}{2}$ , -2.

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ . Choices: $-1$ , 1, $\frac{-1}{e}$ , e.

Find the derivative of $y=\sin(e^{x^{2}})$ with respect to $x$ : $\frac{dy}{dx}$ .

Find the first derivative $\frac{d y}{d x}$ at the point (1,1) for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

Find the integral of $\sec(5x) \, dx$ . Choose the correct answer from the options provided.

Find $\frac{d y}{d x}$ for the equation $e^{y}=6 x+3 y$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ if $x^{2}+y^{2}=1$ . Options: A) $\frac{-1}{y^{3}}$ B) $\frac{x^{2}+y^{2}}{y^{3}}$ C) $\frac{x^{2}-y^{2}}{y^{3}}$ D) $\frac{-x}{y^{2}}$

Find the derivative $f^{\prime}(x)$ if $f(x)=\ln(2 e^{-x} \sin x)$ .

Find $\frac{d y}{d x}$ for $x^{2} y^{4}-2 y=3$ at the point $(2,1)$ . Options: 2, -2, 20, $\frac{1}{2}$ .

Find $\frac{d y}{d x}$ for $x=y \cos (2 x)$ at the point $\left(\frac{\pi}{2}, \frac{-\pi}{2}\right)$ . Options: $-1$ , $\frac{1}{2}$ , 0, 1.

Find $f^{\prime}(e)$ if $f(x)=x^{\ln x}$ . Choices: 1, 0, e, 2.

Evaluate the integral $\int_{0}^{1} x^{2} 2^{x^{3}} d x$ and choose between: a. $\frac{1}{\ln (4)}$ b. $\frac{1}{\ln (8)}$

Find the integral $\int \csc (5 x) d x$ and choose the correct answer from the options provided.

Find the derivative of $f(x)=\ln(2 e^{-x} \sin x)$ . What is $f^{\prime}(x)$ ?

Find $\left(f^{-1}\right)^{\prime}(-3)$ given $f(-3)=5, f(4)=-3, f^{\prime}(-3)=\frac{-1}{4}, f^{\prime}(4)=\frac{1}{2}$ .

Find $\frac{d y}{d x}$ if $e^{y}=6 x+3 y$ . Options: $\frac{6 x}{e^{y}+3}$ , $\frac{6}{e^{y}+3}$ , $\frac{6}{e^{y}+3 y}$ , $\frac{6}{e^{y}-3}$ .

Find $\frac{d y}{d x}$ for $y=\ln \sqrt{\tan x}$ at $x=\frac{\pi}{4}$ . What is the value?

Find $\frac{d y}{d x}$ for the equation $e^{y}=6 x+3 y$ . Options: $\frac{6}{e^{y}+3}$ , $\frac{6 x}{e^{y+3}}$ , $\frac{6}{e^{y}-3}$ , $\frac{6}{e^{y}+3 y}$ .

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

Find the tangent line to the curve $f(x)=e^{2 x}$ at the point $(0,1)$ . Choose the correct equation.

Find $\frac{d y}{d x}$ for $y=\ln \sqrt{\tan x}$ at $x=\frac{\pi}{4}$ .

Find $\frac{d y}{d x}$ at the point (1,1) for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

Calculate the integral $\int_{0}^{\frac{\pi}{6}} \sec (2 x) d x$ and choose the correct answer from the options.

Solve the differential equation: $y^{\prime \prime}+3 y^{\prime}-40 y=6 x^{2}$ .

Bestimmen Sie die Definitionsmenge der Funktionen und die mittlere Änderungsrate in den Intervallen $I_{1}=[-1 ; 0], I_{2}=[0 ; 1], I_{3}=[1 ; 3], I_{4}=[0 ; 3]$ für $f$ .

Calculate the integral $\int_{0}^{1} 9 x^{2} 2^{x^{3}} d x$ . Choose from: (a) $\frac{6}{\ln 2}$ , (b) $\frac{1}{\ln 2}$ , (c) $\ln 2$ , (d) $\frac{3}{\ln 2}$ .

Find the derivative of $x e^{x}-e^{x}$ and choose the correct option.

إذا كانت $\Delta ص = \frac{8هـ - هـ^2}{4}$ ، فما قيمة $ق'(هـ)$ في الفترة $[5, 5+هـ]$ ؟

A conical container (radius $9$ ft, height $36$ ft) has liquid at $31$ ft. Find work to pump it to rim and $5$ ft above rim.

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\frac{8}{9+h}-\frac{8}{9}}{h}$ by direct substitution. Simplify to $\lim _{h \rightarrow 0}(\square)$ .

Find the radius of a loop for a pilot in an aerobatic plane experiencing weightlessness at $310 \mathrm{~m/s}$ with gravity $9.8 \mathrm{~m/s}^2$ . Answer in $\mathrm{km}$ .

Solve the equation $\frac{d R}{d x}=a\left(R^{2}+1\right)$ for $R$ , using constant $C$ .

Calculate $W=\frac{62 \cdot 4 \pi}{48} \cdot \int_{0}^{31} (36 h^{2}-h^{3}) \, dh$ .

Find the derivative of $f(x)=\ln(2 e^{-x} \sin x)$ . What is $f^{\prime}(x)$ ?

33. Eingesperrtes Rechteck: Bestimme die Koordinaten von Punkt P auf $f(x)=1+\frac{4}{x^{2}}$ , um a) Fläche und b) Umfang minimal zu halten.

Bestimmen Sie die Stammfunktion $F(x) = (a \cdot x + b) e^{-0,5x}$ von $f$ durch Koeffizientenvergleich und berechnen Sie $A$ über $I=[0, 7]$ .

Find the derivative of $y=e^{x}(\sin x+\cos x)$ with respect to $x$ .

Find the tangent line to the curve $f(x)=e^{2 x}$ at the point $(0,1)$ . Select the correct equation.

Find $f^{\prime}(e)$ if $f(x)=x^{\ln x}$ . Choices: 0, 1, $e$ , 2.

Find the tangent line to $f(x) = e^{-5x}$ at $x = 0$ and the area of the triangle it forms in the first quadrant.

Solve the differential equation $7x - 8y\sqrt{x^2 + 1} \frac{dy}{dx} = 0$ with $y(0) = 1$ . Find $y =$ .

Find $f'(0)$ if $f(x)=\int_{1}^{e^{2 x}} (t^{2}+\ln t) dt$ . Write a number only Answer:

Bestimme die unbestimmten Integrale: a) $\int 7 \, dx$ b) $\int\left(\frac{1}{3} x^{2}+5 x-3\right) \, dx$

Evaluate the integral $\int \sqrt{x^{2}-25} d x$ and choose the best technique: A, B, C, D, or E?

Find the first derivative $\frac{d y}{d x}$ of $\sin y = x^{3}+y^{2}+8$ at the point $(-2,0)$ .

Find the relative maximum point of $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x+1$ . Use $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ .

Berechnen Sie die Integrale: a) $\int_{0}^{2}(2 x+1) d x$ , b) $\int_{-1}^{3} 0,5 x^{3} d x$ , c) $\int_{0}^{4}\left(x^{2}-\frac{x}{3}\right) d x$ .

Evaluate the limit: $\lim _{x \rightarrow 9} \frac{\sqrt{x}-3}{x-9}$ by simplifying it for direct substitution.