Calculus

Problem 5701

Bestimme die Ableitungsfunktion $f^{\prime}$ für die Funktionen: a) $f(x)=\frac{1}{4} x^{4}-2 x^{2}$ , b) $f(x)=-3 x^{2}+4$ , c) $f(x)=3(x-2)^{2}+x$ , d) $f(x)=a x^{3}+b x^{2}+c x+d$ , e) $f(x)=2 \sqrt{x}$ , f) $f(x)=\frac{4}{x}+1$ .

See Solution

Problem 5702

A ball goes 108 in in the first 0.1s, 102 in in the next, then 6 in less each 0.1s. How long until it falls? Total distance?

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Problem 5703

Berechnen Sie die Integrale: a) $\int_{0}^{4}-x d x$ , b) $\int_{-1}^{1}-2 x d x$ , c) $\int_{-2}^{2}-x^{2} d x$ , d) $\int_{-4}^{-2}-0,5 x d x$ , e) $\int_{-20}^{-10}-1 d x$ , f) $\int_{-1}^{0} d x$ .

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Problem 5704

Leiten Sie die Funktion $f$ mit Produkt- und Kettenregel ab für: a) $f(x)=x \cdot \sin (3 x)$ , b) $f(x)=(2 x-1)^{2} \cdot \sqrt{x}$ , c) $f(x)=3 x^{5} \cdot \cos (2 x)$ , d) $f(x)=3 x \cdot \sin (4 x-1)$ , e) $f(x)=(4-3 x)^{2} \cdot \sin (x)$ , f) $f(x)=0,5 x^{2} \cdot \sqrt{4-x}$ , g) $f(x)=x^{2} \cdot \cos (1-x)$ , h) $f(x)=\sqrt{2 x+3} \cdot x^{2}$ , i) $f(x)=(5 x+2)^{7} \cdot \cos (x)$ .

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Problem 5705

Find the rate of area increase of a square with side $3 \mathrm{~m}$ growing at $0.8 \mathrm{~m/min}$ .

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Problem 5706

Find the derivative of $f(x) = \sqrt{x}$ or $f(x) = x^{\frac{1}{2}}$ .

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Problem 5707

Finde die Tangentengleichung an $f(x)=2 e^{x}$ bei $x_{0}=-1$ .

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Problem 5708

Find the derivative $f'(x)$ of $f(x)=\frac{1}{x^2}$ using the limit definition: $f'(x)=\lim_{h \to 0} \frac{\frac{1}{(x+h)^{2}}-\frac{1}{x^{2}}}{h}$ .

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Problem 5709

Encuentra $T(t)$ , $N(t)$ , $B(t)$ y $\dot{K}(t)$ para $r(t)=\langle 5t-2, e^{-t}, 3t \rangle$ .

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Problem 5710

Find the absolute extrema of $f(x)=-4 x^{3}-54 x^{2}-216 x$ on $[-8,3]$ . Provide your answer as $(x, f(x))$ .

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Problem 5711

Find the absolute extrema of $f(x)=6 x^{3}-99 x^{2}+540 x$ on $[-4,8]$ . Provide $(x, f(x))$ pairs.

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Problem 5712

Find the absolute extrema of $f(x)=5x^{2}-20x$ on $[0,8]$ . Give your answer as an ordered pair $(x, f(x))$ .

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Problem 5713

Find $\lim _{x \rightarrow 1^{-}} \frac{f(x)-f(1)}{x-1}$ for the function $f$ on $[-1,5]$ with given segments and $y=x^{3}$ .

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Problem 5714

Find the absolute extrema of $f(x)=2 x^{3}-216 x$ on $[-8,7]$ . Provide your answer as $(x, f(x))$ .

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Problem 5715

Find the derivative of $f(x)=2x^{3}-216x$ .

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Problem 5716

Find the derivative of the function $f(x) = 5x^2 - 20x$ .

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Problem 5717

Find the absolute extrema of the function $f(x)=-x^{3}-4$ on the interval $[-7,7]$ . Provide as $(x, f(x))$ .

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Problem 5718

Find characteristics of the function $f(x)=(1-x^{2}) \cdot e^{-\frac{1}{2} x^{2}}$ , like its derivative, integral, and extrema.

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Problem 5719

Find the absolute extrema of $f(x)=4 x^{3}-12 x^{2}-288 x$ on $[-7,7]$ as an ordered pair $(x, f(x))$ .

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Problem 5720

Find the absolute extrema of $f(x)=5 x^{2}+10 x-8$ on $[-6,3]$ . Provide your answer as an ordered pair $(x, f(x))$ .

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Problem 5721

A volleyball is spiked from 10 ft with an initial speed of -55 ft/s. How much time do players have to hit it before it lands?

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Problem 5722

Find the absolute extrema of $f(x)=2 x^{3}-216 x$ on $[-8,7]$ by evaluating $f(-8)$ , $f(7)$ , $f(6)$ , and $f(-6)$ .

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Problem 5723

Find the absolute extrema of $f(x)=8 x+\frac{648}{x}$ on [5, 19]. Provide $(x, f(x))$ rounded to 3 decimal places.

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Problem 5724

Find the limit: $\lim _{x \rightarrow \infty} \frac{20 x^{4}-16 x}{35 x^{3}-20 x^{4}+7-4 x}$ . State DNE if infinite.

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Problem 5725

Find the absolute extrema of $f(x)=6 x^{3}-54 x^{2}+144 x$ on $[-2,7]$ . Provide as $(x, f(x))$ pairs.

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Problem 5726

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{3}-41}}{10 x+9+3 x^{2}}$ . State DNE if infinite.

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Problem 5727

Find the limit: $\lim _{x \rightarrow \infty} \frac{63 x^{4}+14 x}{-12 x^{2}+27 x^{4}-4}$ . If infinite, state DNE.

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Problem 5728

Find the value of $x$ that minimizes the average cost function for $C(x)=50000+1.6x+20x^{2}$ , where $1 \leq x \leq 273$ .

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Problem 5729

Find the limit: $\lim _{x \rightarrow \infty} \frac{20 x^{4}-16 x}{35 x^{3}-20 x^{4}+7-4 x}$ . If infinite, state DNE.

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Problem 5730

Find the absolute extrema of $f(x)=-3 x^{3}+81 x$ on $[-5,4]$ . Provide your answer as $(x, f(x))$ .

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Problem 5731

Bestimme die Wendepunkte der Funktion $f(x)=\left(1-x^{2}\right) \cdot e^{-\frac{1}{2} x^{2}}$ .

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Problem 5732

Find the tangent lines at $x=2$ for: a. $y=f(x)+g(x)$ b. $y=f(x)-3g(x)$ c. $y=2f(x)$ Given tangents: $y=5x+1$ for $f$ and $y=4x-2$ for $g$ .

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Problem 5733

Untersuchen Sie die Funktion $f(x)=\left(1-x^{2}\right) \cdot e^{-\frac{1}{2} x^{2}}$ auf Maxima, Minima, Grenzwerte, Nullstellen, Ableitungen und Integrale.

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Problem 5734

Find $\lim _{x \rightarrow 2} f(x)$ for the piecewise function $f(x) = \{ 2 + 2x \text{ if } x \geq 2, 1 + x^2 \text{ if } x < 2 \}$ .

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Problem 5735

Find the absolute extrema of $f(x)=3 x^{3}-36 x$ on $[-3,7]$ . Provide as $(x, f(x))$ .

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Problem 5736

Calculate the integral of the function $f(x) = x e^{-x^2}$ .

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Problem 5737

Calculate the derivative of $f(x) = e^{-x^{2}} - 2x^{2}e^{-x^{2}}$ .

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Problem 5738

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the function $y=3 \sin x+x^{2} \cos x$ .

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Problem 5739

Identify where $f^{\prime}$ is undefined in $[-4,6]$ and graph the step function for the derivative of $f$ .

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Problem 5740

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

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Problem 5741

Find the missing parts of the derivative for $h(t)=(-4 t^{2}+2 t)(-9 t^{3}-8 t^{2}-t)$ .

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Problem 5742

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{x^{2}}{x^{3}+18}$ .

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Problem 5743

Set up an integral for the volume of the solid formed by rotating the area between $y=\sqrt{x}$ , $y=0$ , $x=4$ about $x=9$ .

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Problem 5744

Find the volume $V$ of the solid formed by rotating the area between $3x=y^2$ , $x=0$ , $y=6$ around the $y$ -axis.

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Problem 5745

Für die Funktion $f(x)=b^{x}$ mit $f^{\prime}(0)=1,3$ : a) Wie hängt $f^{\prime}$ mit $f$ zusammen? b) Finde $b$ und den Term für $f^{\prime}$ .

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Problem 5746

Ein Fluss verläuft in einer Senke, beschrieben durch $f(x) = -5 x^{2} e^{x} + 1$ für $x \in [-6; 0]$ .
a) Berechne den Höhenunterschied zwischen den Uferzonen.
b) Bestimme die Breite der Senke einen Meter unterhalb der Wasseroberfläche.
c) Deute $f(x+3)=f(x)$ und finde eine Lösung.
d) Leite $f^{\prime}(x)$ aus $f(x)$ ab.
e) Berechne die Wassertiefe an der tiefsten Stelle.

See Solution

Problem 5747

Find the missing part of the derivative for $k(x)=(-1-4 x^{2})^{6}$ : $k^{\prime}(x)=6(-1-4 x^{2})^{5} \cdot ?$

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Problem 5748

Find the missing part of the derivative for $k(x)=\left(-1-4 x^{2}\right)^{6}$ : $k^{\prime}(x)=\left(6\left(-1-4 x^{2}\right)^{5}\right) \cdot ?$

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Problem 5749

Does the curve $y=2 x^{2}-13 x+5$ have a tangent with slope -1? Find the equation of the tangent line and point of tangency.

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Problem 5750

Find the correct derivative of $f(x)=(\sqrt{5 x}+4)^{6}$ .

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Problem 5751

Find the derivative of $f(x)=9 x^{-10}-9 x^{-3}+12 x^{-2}+10 x^{3}$ . What is $f^{\prime}(x)$ ?

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Problem 5752

Differentiate $f(x)=\frac{(3+2x)^{5}}{(4+2x^{3})^{2}}$ and find $f^{\prime}(1)$ . Provide the exact value.

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Problem 5753

Find the missing part of the derivative for $k(x)=\left(-1-4 x^{2}\right)^{6}$ : $k^{\prime}(x)=\left(6\left(-1-4 x^{2}\right)^{5}\right) \cdot ?$

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Problem 5754

Find $\frac{d y}{d x}$ for the function $y=\frac{x^{4}}{x^{5}+26}$ .

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Problem 5755

Find the missing part of the derivative for $h(t)=(7 t^{3}+2 t^{2})(4-3 t)$ . $h^{\prime}(t)=(\text { click for List }-3 t)+(4-3 \text { click for List })(-3)$

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Problem 5756

A firm has a cost function $T C(q)=50,000+25 q+0.001 q^{2}$ . Find fixed cost $F C$ , marginal cost $M C$ , and $M C$ at $q=100$ and $q=10,000$ .

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Problem 5757

Find the derivative of the function $g(x)=(3x+7)(5x+1)$ . What is $g'(x)$ ?

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Problem 5758

Differentiate $f(x)=\frac{8 x^{2}+8 x+7}{6 x^{2}+5 x+3}$ and find $f^{\prime}(0)$ . Enter an exact value.

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Problem 5759

Ein Fluss fließt in einer Senke, modelliert durch $f(x)=-5 x^{2} e^{x}+1$ für $x \in[-6 ; 0]$ .
Aufgaben: a) Bestimme den Höhenunterschied zwischen den Uferzonen. b) Wie breit ist die Senke 1 m unterhalb der Wasseroberfläche? c) Deute $f(x+3)=f(x)$ und finde eine Lösung. d) Leite $f^{\prime}(x)$ aus $f(x)$ ab. e) Berechne die Wassertiefe an der tiefsten Stelle.

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Problem 5760

Leiten Sie die folgenden Funktionen ab und vereinfachen Sie die Ergebnisse: a) $f(x)=(x+2)^{4}$ b) $f(x)=(8x+2)^{3}$ c) $f(x)=\left(\frac{1}{2}-5x\right)^{3}$ d) $f(x)=\frac{1}{4}\left(x^{2}-5\right)^{2}$ e) $f(x)=(8x-7)^{-1}$ f) $f(x)=(5-x)^{-4}$ g) $f(x)=(15x-3)^{-2}$ h) $f(x)=\left(15x-3x^{2}\right)^{-2}$

See Solution

Problem 5761

Find the missing part of the derivative for $k(x)=\left(2-4 x^{3}\right)^{-5}$ : $k^{\prime}(x)=(\square \text { 固国 })\left(-12 x^{2}\right)$

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Problem 5762

Differentiate $f(x)=\frac{(5+3x^{2})^{5}}{(3-5x^{3})^{2}}$ and find $f^{\prime}(1)$ . What is the exact value?

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Problem 5763

Find the correct derivative of $h(t)=(2 x^{3}+3 x^{2}-4)(-5 x^{2}+2 x-2)$ .

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Problem 5764

Find the correct derivative of $f(x)=(\sqrt{4 x}+2)^{2}$ . Options include $f^{\prime}(x)=2(\sqrt{4 x}+2) \sqrt{4}$ , etc.

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Problem 5765

Find the tangent line equation for $g(x)=(x-1)^{3} \sqrt{x^{2}+15}$ at $x=1$ , in the form $y=m x+b$ .

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Problem 5766

Differentiate $f(x)=\frac{(2+4 x)^{3}}{(-3+x^{3})^{5}}$ and find $f^{\prime}(1)$ . What is the exact value?

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Problem 5767

Given $f(x)=\frac{x^{2}-x}{x-1}$ , determine the behavior of its derivative $f^{\prime}(x)$ at $x=1$ .

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Problem 5768

Die Geschwindigkeit eines Flugkörpers in den ersten 7 Sekunden ist $v(t)=0,1 t^{2}$ . Bestimmen Sie $v'(t)$ , skizzieren Sie $v$ und $v'$ und interpretieren Sie die Änderungsrate.

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Problem 5769

Find $x$ where the derivative of $y=\frac{(5+35 x)^{2}}{80 x}$ equals zero.

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Problem 5770

Given $f(x)=\frac{x^{2}-x}{x-1}$ , find the correct statement about its derivative $f^{\prime}(x)$ .

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Problem 5771

Produktionsfunktion $P(x)=-0,5 x^{3}+1,5 x^{2}+0,075 x ; D(P)=[0 ; 3,05]$ .
a) Wo wächst die Produktion? b) Wo wächst die Produktion am stärksten? c) Wo wächst die Produktionsmenge progressiv? d) Wo wächst die Produktionsmenge degressiv?

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Problem 5772

Given $f(x)=\frac{x^{2}-x}{x-1}$ , which statement about $f^{\prime}(x)$ is true at $x=1$ ?

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Problem 5773

Bestimmen Sie die Stellen, an denen die Funktion die gegebene Steigung $m$ hat.
a) $f(x)=2 x^{3}$ ; $m=6$ b) $f(x)=\frac{3 x^{4}}{2}$ ; $m=48$ c) $f(x)=\frac{-4}{x}$ ; $m=\frac{1}{9}$ d) $f(x)=\frac{x^{2}}{5}$ ; $m=10$ e) $f(x)=3 x$ ; $m=3$ f) $f(x)=5 x$ ; $m=3$ g) $f(x)=3 \sqrt{x}$ ; $m=6$ h) $f(x)=\frac{2}{x^{2}}$ ; $m=\frac{1}{2}$ i) $f(x)=\frac{9}{x^{3}}$ ; $m=-\frac{1}{3}$

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Problem 5774

Find the tangent line equation $y=m x+b$ for the function $g(x)=x^{2} \sqrt{x^{2}+8}$ at $x=-1$ .

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Problem 5775

Find $x$ where the derivative of $y=\frac{(39+26 x)^{2}}{82 x}$ equals zero.

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Problem 5776

Bestimme $k$ für das Abkühlungsgesetz $T(t)=T_{U}+(T_{0}-T_{u}) e^{k t}$ mit $T_{0}=80^{\circ}C$ , $T_{U}=20^{\circ}C$ nach 30 min.

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Problem 5777

Bestimme die Stellen, an denen die Funktion die Steigung $m$ hat für die folgenden Funktionen:
a) $f(x)=2 x^{3} ; \quad m=6$ b) $f(x)=\frac{3 x^{4}}{2} ; \quad m=48$ c) $f(x)=\frac{-4}{x} ; \quad m=\frac{1}{9}$ d) $f(x)=\frac{x^{2}}{5} ; \quad m=10$ e) $f(x)=3 x ; \quad m=3$ f) $f(x)=5 x ; \quad m=3$ g) $f(x)=3 \sqrt{x} ; \quad m=6$ h) $f(x)=\frac{2}{x^{2}} ; \quad m=\frac{1}{2}$ i) $f(x)=\frac{9}{x^{3}} ; \quad m=-\frac{1}{3}$

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Problem 5778

Bestimme die Integrale mit Flächen von Dreiecken und Rechtecken: a) $\int_{2}^{5} x \, dx$ , b) $\int_{-1}^{1}(2x+1) \, dx$ , c) $\int_{-1}^{2}-2t \, dt$ , d) $\int_{0}^{4}-2 \, dx$ , e) $\int_{-5}^{0}(-t-5) \, dt$ .

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Problem 5779

Find the derivative of the function $f(y)=(\cot (y))^{\frac{2}{9}}$ . What is $f^{\prime}(y)$ ?

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Problem 5780

Given $y=(f(x)+5 x^{2})^{4}$ , with $f(-1)=-4$ and $\frac{d y}{d x}=3$ at $x=-1$ , find $f^{\prime}(-1$ . Answer exactly. $f^{\prime}(-1)=$

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Problem 5781

Bestimmen Sie die Ableitung von $f(x)$ an den gegebenen Stellen $x_{0}$ für die Funktionen a) bis f).

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Problem 5782

Find $f^{\prime}(-1)$ given $y=(f(x)+5 x^{2})^{4}$ , $f(-1)=-4$ , and $\frac{d y}{d x}=3$ at $x=-1$ . Answer: $f^{\prime}(-1)=\square$

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Problem 5783

Bestimme die Ableitung der folgenden Funktionen: a) $f(s)=s^{2}-3sy-2$ , b) $f(x)=ax^{3}+bx^{2}$ , c) $f(t)=k-t$ , d) $t(x)=mx+b$ , e) $s(t)=\frac{g}{2}t^{2}-0,3t+2$ , f) $f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e$ .

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Problem 5784

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{x^{3}}{x^{4}+1}$ .

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Problem 5785

Calculate the limit as $x$ approaches 3 from the left: $\lim _{x \rightarrow 3^{-}}\left(\frac{x}{x-3}-\frac{3}{x^{2}-9}\right)$ .

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Problem 5786

Find the intervals where the function $f(x)=\left\{\begin{array}{cc}2 x-2 & \text { if } x<1 \\ x^{2} & \text { if } 1 \leq x \leq 2 \\ 2 x^{3}-12 & \text { if } x \geq 2\end{array}\right.$ is continuous.

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Problem 5787

Determine where the function $f(x)=\left\{\begin{array}{cc}\frac{x^{3}+1}{x+1} & \text { if }-1<x<2 \\ 4 & \text { if } x=-1 \\ \ln (x-1) & \text { if } x \geq 2\end{array}\right.$ is continuous.

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Problem 5788

Bestimme die Tangentengleichung an den Wendepunkt $W(3,33 \mid 1,48)$ der Funktion $f(x)=-0,02 \cdot x^{3}+0,2 \cdot x^{2}$ .

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Problem 5789

Find the missing part of the derivative for $h(t)=\left(t^{2}+9\right)\left(-5 t^{3}-7 t^{2}+8\right)$ .

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Problem 5790

Finde die Ableitungen der Funktionen und analysiere die Funktion $f(x)=-\frac{1}{2} x^{3}+1.6 x$ hinsichtlich:
1. Definitionsbereich
2. Symmetrie
3. Achsenschnittpunkte
4. Extrempunkte
5. Wendepunkt

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Problem 5791

Find the missing part of the derivative for $k(x)=\left(-5+4 x^{3}\right)^{-4}$ : $k^{\prime}(x)=(\square \text { 固园 })\left(12 x^{2}\right)$

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Problem 5792

Find the correct derivative of the function $h(t)=(-3 x^{3}+x^{2}+2 x+2)(4+5 x)$ .

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Problem 5793

Find the tangent line equation $y=m x+b$ for the function $g(x)=(x+2)^{3} \sqrt{x^{2}-3}$ at $x=-2$ .

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Problem 5794

Find the value of $x$ where $\frac{d y}{d x}=0$ for $y=\frac{16}{x^{2}}+\frac{128}{(3-x)^{2}}$ .

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Problem 5795

Differentiate and simplify these functions:
a) $f(x)=\frac{1}{(x-1)^{2}}$ , b) $f(x)=\frac{1}{(3x-1)^{2}}$ , c) $f(x)=\frac{3}{(x-1)^{2}}$ , d) $f(x)=\frac{1}{3(x-1)^{2}}$ , e) $f(x)=e^{7x}$ , f) $f(x)=e^{2x^{2}}+x$ , g) $f(x)=4e^{3-4x}$ , h) $f(x)=\frac{1}{2}e^{\frac{1}{3}x+2}$ .

See Solution

Problem 5796

a) Vervollständigen Sie die Ableitungen: $f^{\prime}(x)=\triangle e^{x}$ und $g^{\prime}(x)=\square(1-3 x)^{\triangle}$ . b) Finden Sie den Fehler in den Ableitungen: $f^{\prime}(x)=4(5-2 x)^{3}$ und $g^{\prime}(x)=-4 e^{1-x}$ .

See Solution

Problem 5797

3 a) Vervollständigen Sie: $f(x)=2 e^{x} ; f^{\prime}(x)=\triangle e^{x} ; g(x)=0,5(1-3 x)^{4} ; g^{\prime}(x)=\square(1-3 x)^{\triangle}$ b) Wo ist der Fehler? $f(x)=(5-2 x)^{4} ; f^{\prime}(x)=4(5-2 x)^{3} ; g(x)=4 e^{1-x} ; g^{\prime}(x)=-4 e^{1-x}$

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Problem 5798

Berechne die Flächeninhaltsfunktion $\mathrm{A}_{0}(\mathrm{x})$ für $f(x)=\frac{1}{3} x^{2}$ . Bestimme die Flächeninhalte für $[0;1]$ , $[0;2]$ und $[1;2]$ .

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Problem 5799

Approximate sugar production when price is 28 cents, using $f(30) = 6700$ and $f'(30) = 50$ with linearization.

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Problem 5800

Monopoly profit maximization: Given $TC(q)=q+0.02q^{2}$ and $q+20p=300$ , find (a) inverse demand, (b) profit function, (c) optimal $q_{m}$ , max profit, (d) price.

See Solution

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Calculus

Problem 5701

Problem 5702

A ball goes 108 in in the first 0.1s, 102 in in the next, then 6 in less each 0.1s. How long until it falls? Total distance?

Problem 5703

Problem 5704

Problem 5705

Find the rate of area increase of a square with side 3 m3 \mathrm{~m}3 m growing at 0.8 m/min0.8 \mathrm{~m/min}0.8 m/min.

Problem 5706

Find the derivative of f(x)=xf(x) = \sqrt{x}f(x)=x​ or f(x)=x12f(x) = x^{\frac{1}{2}}f(x)=x21​.

Problem 5707

Finde die Tangentengleichung an f(x)=2exf(x)=2 e^{x}f(x)=2ex bei x0=−1x_{0}=-1x0​=−1.

Problem 5708

Find the derivative f′(x)f'(x)f′(x) of f(x)=1x2f(x)=\frac{1}{x^2}f(x)=x21​ using the limit definition: f′(x)=lim⁡h→01(x+h)2−1x2hf'(x)=\lim_{h \to 0} \frac{\frac{1}{(x+h)^{2}}-\frac{1}{x^{2}}}{h}f′(x)=limh→0​h(x+h)21​−x21​​.

Problem 5709

Encuentra T(t)T(t)T(t), N(t)N(t)N(t), B(t)B(t)B(t) y K˙(t)\dot{K}(t)K˙(t) para r(t)=⟨5t−2,e−t,3t⟩r(t)=\langle 5t-2, e^{-t}, 3t \rangler(t)=⟨5t−2,e−t,3t⟩.

Problem 5710

Find the absolute extrema of f(x)=−4x3−54x2−216xf(x)=-4 x^{3}-54 x^{2}-216 xf(x)=−4x3−54x2−216x on [−8,3][-8,3][−8,3]. Provide your answer as (x,f(x))(x, f(x))(x,f(x)).

Problem 5711

Find the absolute extrema of f(x)=6x3−99x2+540xf(x)=6 x^{3}-99 x^{2}+540 xf(x)=6x3−99x2+540x on [−4,8][-4,8][−4,8]. Provide (x,f(x))(x, f(x))(x,f(x)) pairs.

Problem 5712

Find the absolute extrema of f(x)=5x2−20xf(x)=5x^{2}-20xf(x)=5x2−20x on [0,8][0,8][0,8]. Give your answer as an ordered pair (x,f(x))(x, f(x))(x,f(x)).

Problem 5713

Find lim⁡x→1−f(x)−f(1)x−1\lim _{x \rightarrow 1^{-}} \frac{f(x)-f(1)}{x-1}limx→1−​x−1f(x)−f(1)​ for the function fff on [−1,5][-1,5][−1,5] with given segments and y=x3y=x^{3}y=x3.

Problem 5714

Find the absolute extrema of f(x)=2x3−216xf(x)=2 x^{3}-216 xf(x)=2x3−216x on [−8,7][-8,7][−8,7]. Provide your answer as (x,f(x))(x, f(x))(x,f(x)).

Problem 5715

Find the derivative of f(x)=2x3−216xf(x)=2x^{3}-216xf(x)=2x3−216x.

Problem 5716

Find the derivative of the function f(x)=5x2−20xf(x) = 5x^2 - 20xf(x)=5x2−20x.

Problem 5717

Find the absolute extrema of the function f(x)=−x3−4f(x)=-x^{3}-4f(x)=−x3−4 on the interval [−7,7][-7,7][−7,7]. Provide as (x,f(x))(x, f(x))(x,f(x)).

Problem 5718

Find characteristics of the function f(x)=(1−x2)⋅e−12x2f(x)=(1-x^{2}) \cdot e^{-\frac{1}{2} x^{2}}f(x)=(1−x2)⋅e−21​x2, like its derivative, integral, and extrema.

Problem 5719

Find the absolute extrema of f(x)=4x3−12x2−288xf(x)=4 x^{3}-12 x^{2}-288 xf(x)=4x3−12x2−288x on [−7,7][-7,7][−7,7] as an ordered pair (x,f(x))(x, f(x))(x,f(x)).

Problem 5720

Find the absolute extrema of f(x)=5x2+10x−8f(x)=5 x^{2}+10 x-8f(x)=5x2+10x−8 on [−6,3][-6,3][−6,3]. Provide your answer as an ordered pair (x,f(x))(x, f(x))(x,f(x)).

Problem 5721

A volleyball is spiked from 10 ft with an initial speed of -55 ft/s. How much time do players have to hit it before it lands?

Problem 5722

Find the absolute extrema of f(x)=2x3−216xf(x)=2 x^{3}-216 xf(x)=2x3−216x on [−8,7][-8,7][−8,7] by evaluating f(−8)f(-8)f(−8), f(7)f(7)f(7), f(6)f(6)f(6), and f(−6)f(-6)f(−6).

Problem 5723

Find the absolute extrema of f(x)=8x+648xf(x)=8 x+\frac{648}{x}f(x)=8x+x648​ on [5, 19]. Provide (x,f(x))(x, f(x))(x,f(x)) rounded to 3 decimal places.

Problem 5724

Find the limit: lim⁡x→∞20x4−16x35x3−20x4+7−4x\lim _{x \rightarrow \infty} \frac{20 x^{4}-16 x}{35 x^{3}-20 x^{4}+7-4 x}x→∞lim​35x3−20x4+7−4x20x4−16x​. State DNE if infinite.

Problem 5725

Find the absolute extrema of f(x)=6x3−54x2+144xf(x)=6 x^{3}-54 x^{2}+144 xf(x)=6x3−54x2+144x on [−2,7][-2,7][−2,7]. Provide as (x,f(x))(x, f(x))(x,f(x)) pairs.

Problem 5726

Find the limit: lim⁡x→∞9x3−4110x+9+3x2\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{3}-41}}{10 x+9+3 x^{2}}x→∞lim​10x+9+3x29x3−41​​. State DNE if infinite.

Problem 5727

Find the limit: lim⁡x→∞63x4+14x−12x2+27x4−4\lim _{x \rightarrow \infty} \frac{63 x^{4}+14 x}{-12 x^{2}+27 x^{4}-4}x→∞lim​−12x2+27x4−463x4+14x​. If infinite, state DNE.

Problem 5728

Find the value of xxx that minimizes the average cost function for C(x)=50000+1.6x+20x2C(x)=50000+1.6x+20x^{2}C(x)=50000+1.6x+20x2, where 1≤x≤2731 \leq x \leq 2731≤x≤273.

Problem 5729

Find the limit: lim⁡x→∞20x4−16x35x3−20x4+7−4x\lim _{x \rightarrow \infty} \frac{20 x^{4}-16 x}{35 x^{3}-20 x^{4}+7-4 x}x→∞lim​35x3−20x4+7−4x20x4−16x​. If infinite, state DNE.

Problem 5730

Find the absolute extrema of f(x)=−3x3+81xf(x)=-3 x^{3}+81 xf(x)=−3x3+81x on [−5,4][-5,4][−5,4]. Provide your answer as (x,f(x))(x, f(x))(x,f(x)).

Problem 5731

Bestimme die Wendepunkte der Funktion f(x)=(1−x2)⋅e−12x2f(x)=\left(1-x^{2}\right) \cdot e^{-\frac{1}{2} x^{2}}f(x)=(1−x2)⋅e−21​x2.

Problem 5732

Find the tangent lines at x=2x=2x=2 for: a. y=f(x)+g(x)y=f(x)+g(x)y=f(x)+g(x) b. y=f(x)−3g(x)y=f(x)-3g(x)y=f(x)−3g(x) c. y=2f(x)y=2f(x)y=2f(x) Given tangents: y=5x+1y=5x+1y=5x+1 for fff and y=4x−2y=4x-2y=4x−2 for ggg.

Problem 5733

Untersuchen Sie die Funktion f(x)=(1−x2)⋅e−12x2f(x)=\left(1-x^{2}\right) \cdot e^{-\frac{1}{2} x^{2}}f(x)=(1−x2)⋅e−21​x2 auf Maxima, Minima, Grenzwerte, Nullstellen, Ableitungen und Integrale.

Problem 5734

Find lim⁡x→2f(x)\lim _{x \rightarrow 2} f(x)limx→2​f(x) for the piecewise function f(x)={2+2x if x≥2,1+x2 if x<2}f(x) = \{ 2 + 2x \text{ if } x \geq 2, 1 + x^2 \text{ if } x < 2 \}f(x)={2+2x if x≥2,1+x2 if x<2}.

Problem 5735

Find the absolute extrema of f(x)=3x3−36xf(x)=3 x^{3}-36 xf(x)=3x3−36x on [−3,7][-3,7][−3,7]. Provide as (x,f(x))(x, f(x))(x,f(x)).

Problem 5736

Calculate the integral of the function f(x)=xe−x2f(x) = x e^{-x^2}f(x)=xe−x2.

Problem 5737

Calculate the derivative of f(x)=e−x2−2x2e−x2f(x) = e^{-x^{2}} - 2x^{2}e^{-x^{2}}f(x)=e−x2−2x2e−x2.

Problem 5738

Find the second derivative d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for the function y=3sin⁡x+x2cos⁡xy=3 \sin x+x^{2} \cos xy=3sinx+x2cosx.

Problem 5739

Identify where f′f^{\prime}f′ is undefined in [−4,6][-4,6][−4,6] and graph the step function for the derivative of fff.

Problem 5740

Find the derivative of f(x)=x2f(x)=x^{2}f(x)=x2. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 5741

Find the missing parts of the derivative for h(t)=(−4t2+2t)(−9t3−8t2−t)h(t)=(-4 t^{2}+2 t)(-9 t^{3}-8 t^{2}-t)h(t)=(−4t2+2t)(−9t3−8t2−t).

Find the rate of area increase of a square with side $3 \mathrm{~m}$ growing at $0.8 \mathrm{~m/min}$ .

Find the derivative of $f(x) = \sqrt{x}$ or $f(x) = x^{\frac{1}{2}}$ .

Finde die Tangentengleichung an $f(x)=2 e^{x}$ bei $x_{0}=-1$ .

Find the derivative $f'(x)$ of $f(x)=\frac{1}{x^2}$ using the limit definition: $f'(x)=\lim_{h \to 0} \frac{\frac{1}{(x+h)^{2}}-\frac{1}{x^{2}}}{h}$ .

Encuentra $T(t)$ , $N(t)$ , $B(t)$ y $\dot{K}(t)$ para $r(t)=\langle 5t-2, e^{-t}, 3t \rangle$ .

Find the absolute extrema of $f(x)=-4 x^{3}-54 x^{2}-216 x$ on $[-8,3]$ . Provide your answer as $(x, f(x))$ .

Find the absolute extrema of $f(x)=6 x^{3}-99 x^{2}+540 x$ on $[-4,8]$ . Provide $(x, f(x))$ pairs.

Find the absolute extrema of $f(x)=5x^{2}-20x$ on $[0,8]$ . Give your answer as an ordered pair $(x, f(x))$ .

Find $\lim _{x \rightarrow 1^{-}} \frac{f(x)-f(1)}{x-1}$ for the function $f$ on $[-1,5]$ with given segments and $y=x^{3}$ .

Find the absolute extrema of $f(x)=2 x^{3}-216 x$ on $[-8,7]$ . Provide your answer as $(x, f(x))$ .

Find the derivative of $f(x)=2x^{3}-216x$ .

Find the derivative of the function $f(x) = 5x^2 - 20x$ .

Find the absolute extrema of the function $f(x)=-x^{3}-4$ on the interval $[-7,7]$ . Provide as $(x, f(x))$ .

Find characteristics of the function $f(x)=(1-x^{2}) \cdot e^{-\frac{1}{2} x^{2}}$ , like its derivative, integral, and extrema.

Find the absolute extrema of $f(x)=4 x^{3}-12 x^{2}-288 x$ on $[-7,7]$ as an ordered pair $(x, f(x))$ .

Find the absolute extrema of $f(x)=5 x^{2}+10 x-8$ on $[-6,3]$ . Provide your answer as an ordered pair $(x, f(x))$ .

Find the absolute extrema of $f(x)=2 x^{3}-216 x$ on $[-8,7]$ by evaluating $f(-8)$ , $f(7)$ , $f(6)$ , and $f(-6)$ .

Find the absolute extrema of $f(x)=8 x+\frac{648}{x}$ on [5, 19]. Provide $(x, f(x))$ rounded to 3 decimal places.

Find the limit: $\lim _{x \rightarrow \infty} \frac{20 x^{4}-16 x}{35 x^{3}-20 x^{4}+7-4 x}$ . State DNE if infinite.

Find the absolute extrema of $f(x)=6 x^{3}-54 x^{2}+144 x$ on $[-2,7]$ . Provide as $(x, f(x))$ pairs.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{3}-41}}{10 x+9+3 x^{2}}$ . State DNE if infinite.

Find the limit: $\lim _{x \rightarrow \infty} \frac{63 x^{4}+14 x}{-12 x^{2}+27 x^{4}-4}$ . If infinite, state DNE.

Find the value of $x$ that minimizes the average cost function for $C(x)=50000+1.6x+20x^{2}$ , where $1 \leq x \leq 273$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{20 x^{4}-16 x}{35 x^{3}-20 x^{4}+7-4 x}$ . If infinite, state DNE.

Find the absolute extrema of $f(x)=-3 x^{3}+81 x$ on $[-5,4]$ . Provide your answer as $(x, f(x))$ .

Bestimme die Wendepunkte der Funktion $f(x)=\left(1-x^{2}\right) \cdot e^{-\frac{1}{2} x^{2}}$ .

Find the tangent lines at $x=2$ for: a. $y=f(x)+g(x)$ b. $y=f(x)-3g(x)$ c. $y=2f(x)$ Given tangents: $y=5x+1$ for $f$ and $y=4x-2$ for $g$ .

Untersuchen Sie die Funktion $f(x)=\left(1-x^{2}\right) \cdot e^{-\frac{1}{2} x^{2}}$ auf Maxima, Minima, Grenzwerte, Nullstellen, Ableitungen und Integrale.

Find $\lim _{x \rightarrow 2} f(x)$ for the piecewise function $f(x) = \{ 2 + 2x \text{ if } x \geq 2, 1 + x^2 \text{ if } x < 2 \}$ .

Find the absolute extrema of $f(x)=3 x^{3}-36 x$ on $[-3,7]$ . Provide as $(x, f(x))$ .

Calculate the integral of the function $f(x) = x e^{-x^2}$ .

Calculate the derivative of $f(x) = e^{-x^{2}} - 2x^{2}e^{-x^{2}}$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the function $y=3 \sin x+x^{2} \cos x$ .

Identify where $f^{\prime}$ is undefined in $[-4,6]$ and graph the step function for the derivative of $f$ .

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

Find the missing parts of the derivative for $h(t)=(-4 t^{2}+2 t)(-9 t^{3}-8 t^{2}-t)$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{x^{2}}{x^{3}+18}$ .

Set up an integral for the volume of the solid formed by rotating the area between $y=\sqrt{x}$ , $y=0$ , $x=4$ about $x=9$ .

Find the volume $V$ of the solid formed by rotating the area between $3x=y^2$ , $x=0$ , $y=6$ around the $y$ -axis.

Für die Funktion $f(x)=b^{x}$ mit $f^{\prime}(0)=1,3$ : a) Wie hängt $f^{\prime}$ mit $f$ zusammen? b) Finde $b$ und den Term für $f^{\prime}$ .

Find the missing part of the derivative for $k(x)=(-1-4 x^{2})^{6}$ : $k^{\prime}(x)=6(-1-4 x^{2})^{5} \cdot ?$

Find the missing part of the derivative for $k(x)=\left(-1-4 x^{2}\right)^{6}$ : $k^{\prime}(x)=\left(6\left(-1-4 x^{2}\right)^{5}\right) \cdot ?$

Does the curve $y=2 x^{2}-13 x+5$ have a tangent with slope -1? Find the equation of the tangent line and point of tangency.

Find the correct derivative of $f(x)=(\sqrt{5 x}+4)^{6}$ .

Find the derivative of $f(x)=9 x^{-10}-9 x^{-3}+12 x^{-2}+10 x^{3}$ . What is $f^{\prime}(x)$ ?

Differentiate $f(x)=\frac{(3+2x)^{5}}{(4+2x^{3})^{2}}$ and find $f^{\prime}(1)$ . Provide the exact value.

Find the missing part of the derivative for $k(x)=\left(-1-4 x^{2}\right)^{6}$ : $k^{\prime}(x)=\left(6\left(-1-4 x^{2}\right)^{5}\right) \cdot ?$

Find $\frac{d y}{d x}$ for the function $y=\frac{x^{4}}{x^{5}+26}$ .

A firm has a cost function $T C(q)=50,000+25 q+0.001 q^{2}$ . Find fixed cost $F C$ , marginal cost $M C$ , and $M C$ at $q=100$ and $q=10,000$ .

Find the derivative of the function $g(x)=(3x+7)(5x+1)$ . What is $g'(x)$ ?

Differentiate $f(x)=\frac{8 x^{2}+8 x+7}{6 x^{2}+5 x+3}$ and find $f^{\prime}(0)$ . Enter an exact value.

Find the missing part of the derivative for $k(x)=\left(2-4 x^{3}\right)^{-5}$ : $k^{\prime}(x)=(\square \text { 固国 })\left(-12 x^{2}\right)$

Differentiate $f(x)=\frac{(5+3x^{2})^{5}}{(3-5x^{3})^{2}}$ and find $f^{\prime}(1)$ . What is the exact value?

Find the correct derivative of $h(t)=(2 x^{3}+3 x^{2}-4)(-5 x^{2}+2 x-2)$ .

Find the correct derivative of $f(x)=(\sqrt{4 x}+2)^{2}$ . Options include $f^{\prime}(x)=2(\sqrt{4 x}+2) \sqrt{4}$ , etc.

Find the tangent line equation for $g(x)=(x-1)^{3} \sqrt{x^{2}+15}$ at $x=1$ , in the form $y=m x+b$ .

Differentiate $f(x)=\frac{(2+4 x)^{3}}{(-3+x^{3})^{5}}$ and find $f^{\prime}(1)$ . What is the exact value?

Given $f(x)=\frac{x^{2}-x}{x-1}$ , determine the behavior of its derivative $f^{\prime}(x)$ at $x=1$ .

Die Geschwindigkeit eines Flugkörpers in den ersten 7 Sekunden ist $v(t)=0,1 t^{2}$ . Bestimmen Sie $v'(t)$ , skizzieren Sie $v$ und $v'$ und interpretieren Sie die Änderungsrate.

Find $x$ where the derivative of $y=\frac{(5+35 x)^{2}}{80 x}$ equals zero.

Given $f(x)=\frac{x^{2}-x}{x-1}$ , find the correct statement about its derivative $f^{\prime}(x)$ .

Given $f(x)=\frac{x^{2}-x}{x-1}$ , which statement about $f^{\prime}(x)$ is true at $x=1$ ?

Find the tangent line equation $y=m x+b$ for the function $g(x)=x^{2} \sqrt{x^{2}+8}$ at $x=-1$ .