Calculus

Problem 7001

Find $\frac{d V}{d t}$ for a sphere where $V=\frac{4}{3} \pi r^{3}$ in terms of $\frac{d r}{d t}$ . Choose the correct option.

See Solution

Problem 7002

Show how $\frac{d G}{d t}$ relates to $\frac{d M}{d t}$ by differentiating $G=\mathrm{CM}^{0.7}$ with respect to $t$ . $\frac{d G}{d t}=\square \frac{d M}{d t}$

See Solution

Problem 7003

A study shows tree growth rate $G$ depends on body mass $M$ as $G=CM^{0.7}$ . Differentiate to find $\frac{d G}{d t}$ . $\frac{d G}{d t}=\square \frac{d M}{d t}$

See Solution

Problem 7004

Differentiate $G=C M^{0.7}$ with respect to $t$ and express $\frac{d G}{d t}$ in terms of $\frac{d M}{d t}$ .

See Solution

Problem 7005

Find $\frac{d A}{d t}$ for a circle with area $A=\pi r^{2}$ in terms of $\frac{d r}{d t}$ . Choose A, B, C, or D.

See Solution

Problem 7006

A study shows tree growth rate $\mathrm{G}$ depends on mass $\mathrm{M}$ : $G=CM^{0.7}$ . Find $\frac{d G}{d t}$ and relate to $\frac{d M}{d t}$ .

See Solution

Problem 7007

Rearrange the equation for fractional growth rates:
$\frac{d G}{d t}=0.7 \mathrm{CM}^{0.7} \cdot \frac{1}{M} \frac{d M}{d t}$
Show how $\frac{1}{G} \frac{d G}{d t}$ relates to $\frac{1}{M} \frac{d M}{d t}$ .

See Solution

Problem 7008

Find the slope of the tangent line to $y=\ln (2 \cos x)$ at $x=\frac{\pi}{4}$ . Give your answer in simplest form.

See Solution

Problem 7009

Finde die Ableitungsfunktion von $f(x)=0,03 x^{4}-4 x^{3}+2 x$ .

See Solution

Problem 7010

Find the integral of the function: $\int 60 x^{2} e^{5 x^{3}} d x$

See Solution

Problem 7011

Bestimme die Ableitungsfunktion von $f(x)=0,03 x^{4}-4 x^{3}+2 x$ .

See Solution

Problem 7012

Find the value of $f^{\prime}(3)$ for the function $f(x)=x^{2} \ln (2 x)$ .

See Solution

Problem 7013

Bestimme die Ableitung von $f(x)=6 x^{5}+5 x^{3}$ mit Faktor- und Summenregel.

See Solution

Problem 7014

Bestimme die Ableitung von $f(x)=-3 x^{5}-2 x+1$ .

See Solution

Problem 7015

Finde die Ableitung von $f(x)=0,03 x^{5}-3 x^{3}+3 x$ .

See Solution

Problem 7016

Find the limit: $\lim _{x \rightarrow 8}(3+\sqrt[3]{x})(2-6 x^{2}+x^{3})$ . If none, enter DNE.

See Solution

Problem 7017

Bestimme die Ableitung von $f(x)=\frac{1}{3} x^{6}+3 x^{5}+2$ .

See Solution

Problem 7018

Solve the differential equation: $x y' - 2y = x^2$ , for $x > 0$ .

See Solution

Problem 7019

Solve for $y$ in the equation $e^{5 x^{3} y}=\int 60 x^{2} e^{5 x^{3}} d x$ .

See Solution

Problem 7020

Calculate the average velocity of a pebble from $t=2$ for (i) 0.1s, (ii) 0.05s, (iii) 0.01s using $y=235-16t^2$ . Also, find instantaneous velocity at $t=2$ .

See Solution

Problem 7021

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2x^{2}+x-1$ and simplify your answer.

See Solution

Problem 7022

Find the slope and equation of the tangent line to the curve $3xy - 2\pi \cos(y) = 5\pi$ at the point $(1, \pi)$ .

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Problem 7023

Identify where $f$ is discontinuous and check continuity from the right, left, or neither for each $x$ -value.
$f(x)=\left\{\begin{array}{ll} 4^{x} & \text { if } x \leq 1 \\ 5-x & \text { if } 1<x \leq 6 \\ \sqrt{x} & \text { if } x>6 \end{array}\right.$

See Solution

Problem 7024

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{1+81 x^{6}}}{7-x^{3}}$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

See Solution

Problem 7025

Berechne das Integral von $f(x)=2-\frac{x}{3}$ über das Intervall $[-4, 6]$ .

See Solution

Problem 7026

Find the derivative of $\operatorname{SEC} x$ .

See Solution

Problem 7027

Find the limit: $\lim _{x \rightarrow \infty} \frac{4 x^{2}-x+4}{2 x^{2}+5 x-7}$ .

See Solution

Problem 7028

Find the $x$ -values of critical points for $y=x^{3}(6-x)^{6}$ in the range -10 to 10.

See Solution

Problem 7029

Find and simplify the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{2 x^{2} \tan x}{\sec x}$ .

See Solution

Problem 7030

Find the tangent line equation for $y=2 \tan x$ at $\left(\frac{\pi}{4}, 2\right)$ in the form $y=m x+b$ .

See Solution

Problem 7031

Find the derivative of the function $2 x^{2}-\csc (x)+7$ .

See Solution

Problem 7032

Find the tangent line equation for $y=4 \sec x-8 \cos x$ at $\left(\frac{\pi}{3}, 4\right)$ in the form $y=m x+b$ .

See Solution

Problem 7033

Find the derivative of $\operatorname{TAN} x$ .

See Solution

Problem 7034

Find the derivative of $f(x)=-2 \cos x+5 \tan x$ and evaluate $f^{\prime}\left(\frac{2 \pi}{3}\right)$ rounded to the nearest hundredth.

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Problem 7035

Evaluate $f(x)=\frac{6 e^{x}-6-6 x}{x^{2}}$ at $x=1, 0.5, 0.1, 0.05, 0.01, -1, -0.5, -0.1, -0.05, -0.01$ . Find $\lim_{x \rightarrow 0} f(x)$ .

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Problem 7036

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{9+h}-3}{h}$ . If it doesn't exist, write DNE.

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Problem 7037

Find the derivative $\frac{d y}{d x}$ for $y=\frac{8 \sec (x)}{x^{5}}$ . No simplification needed. $\frac{d y}{d x}=$

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Problem 7038

Find the limit: $\lim _{x \rightarrow-1} \frac{2 x^{2}+1}{x^{2}+4 x-5}$ . If it doesn't exist, write DNE.

See Solution

Problem 7039

Differentiate the function using the Product Rule: $f(x)=(6x+3)(x^{3}-4)$ . Find $f'(x)=$ .

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Problem 7040

Differentiate the function using the Quotient Rule: $f(x)=\frac{x}{x^{7}+5}$ , find $f^{\prime \prime}(x)$ .

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Problem 7041

Find the derivative $\frac{d y}{d x}$ for $y=\frac{8 \sec (x)}{x^{10}}$ . No need to simplify. $\frac{d y}{d x}=$

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Problem 7042

Find the derivative $f^{\prime}(x)$ and evaluate it at $c=0$ for $f(x)=(x^{4}+2x)(3x^{3}+5x-3)$ .

See Solution

Problem 7043

A smoothie machine starts with 300g protein in 100L water. After 9 mins of malfunction, find protein left.
(a) Find smoothie volume as a function of time. What are the units? (b) Let $P(t)$ be protein amount; express protein density. (c) Set up a differential equation for protein leaving the tank with units. (d) Solve the equation, find the constant, and determine protein after 9 mins.

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Problem 7044

Calculate the integral $\int_{1}^{2} x^{5} \sqrt{x^{3}-1} \, dx$ .

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Problem 7045

Evaluate the integral $\int_{1}^{2} x^{5} \sqrt{x^{3}-1} \, dx$ .

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Problem 7046

Diana has 80 yards of fencing for a rectangle. Find the area $A$ as a function of width $W$ , max width for largest area, and max area.

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Problem 7047

Find the derivative $f^{\prime}(x)$ of $f(x)=\frac{x+5}{x-4}$ and calculate $f^{\prime}(9)$ .

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Problem 7048

Find the derivative of the function $f(x)=\frac{4-2x-x^{5}}{x^{3}-1}$ . What is $f^{\prime}(x)$ ?

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Problem 7049

Substitute $x=28$ into the rate of change expression: $-\frac{1}{79-28}$ . Round your answer to three decimal places.

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Problem 7050

Given $y^{2}+8x=x^{2}y-7$ and $y(4)=3$ , find $y'(4)$ and the tangent line equation at $(4,3)$ . $y=$

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Problem 7051

Find the volume change rate of a cylinder with radius $\sqrt{t+6}$ and height $\frac{1}{6} \sqrt{t}$ .

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Problem 7052

Find the acceleration of a car from rest at 5, 10, and 20 seconds using $v(t)=\frac{115 t}{8 t+15}$ . Round to three decimal places.

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Problem 7053

Find the second derivative of $f(x)=x^{4}+4x^{3}-7x^{2}-2x$ . What is $f^{\prime \prime}(x)$ ?

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Problem 7054

Find the second derivative of $f(x)=x \cos (x)$ , denoted as $f^{\prime \prime}(x)$ .

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Problem 7055

Find $f^{\prime}(2)$ given $g(2)=2$ , $g^{\prime}(2)=-3$ , $h(2)=-1$ , $h^{\prime}(2)=5$ , and $f(x)=\frac{g(x)}{h(x)}$ .

See Solution

Problem 7056

Find the second derivative of $f(x)=3x+37x^{-2}$ . What is $f^{\prime \prime}(x)$ ?

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Problem 7057

Find the derivative of the function $g(x)=\sqrt{x^{2}-2x+1}$ .

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Problem 7058

Find the derivative $f'(x)$ for $f(x)=(x^{2}+4x+5)^{3}$ and calculate $f'(4)$ .

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Problem 7059

Find $f^{\prime}(x)$ for $f(x)=\sqrt{4 x+6}$ and calculate $f^{\prime}(5)$ .

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Problem 7060

Find the slope of the tangent line to $y=\sin(3x)$ at the origin: $y'(0)=?$

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Problem 7061

Find the growth rate of infected people after 10 days using $I(t)=5500 e^{0.053 t}$ . Round to the nearest tenth.

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Problem 7062

Find the derivative of the function $f(x)=x^{5} e^{7.5 x}$ . What is $f^{\prime}(x)=?$

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Problem 7063

Find the growth rate of infected people after 11 days using $I(t)=3200 e^{0.035 t}$ . Round to the nearest tenth.

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Problem 7064

Find the slope of the tangent line to $y=\sin(3x)$ and $y=\sin(x)$ at the origin. Compare with cycles in $[0, 2\pi]$ .

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Problem 7065

If $h(x)=f(g(x))$ , find $h^{\prime}(1)$ using the values of $f$ and $g$ provided.

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Problem 7066

Find $F'(3)$ and $G'(3)$ given $F(x)=f(f(x))$ , $G(x)=(F(x))^2$ , and $f(3)=4$ , $f(4)=2$ , $f'(4)=15$ , $f'(3)=15$ .

See Solution

Problem 7067

Find the derivative of the function $f(y)=e^{-\sin y+6 \cos y}$ . What is $f^{\prime}(y)$ ?

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Problem 7068

Differentiate $f(w)=\cot(4w^3-9)$ and simplify the result. Find $f'(w)=$

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Problem 7069

Find the derivative of the function $f(x)=\frac{4}{(x^{2}-3x)^{2}}$ and evaluate it at $x=4$ .

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Problem 7070

Find the derivative $f'(4)$ of the function $f(x)=\frac{4}{(x^{2}-3x)^{2}}$ .

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Problem 7071

Find the rate of change of depth at 3 a.m. for the function $D(t)=4 \sin \left(\frac{\pi}{6} t+\frac{5 \pi}{6}\right)+2$ .

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Problem 7072

Find the derivative of $y=\frac{6}{x}+\sqrt{\cos (x)}$ at $(\pi / 2,12 / \pi)$ . Round to three decimal places.

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Problem 7073

Find the derivative of $-8 \sin^{2}(3x^{8})$ using the chain rule twice.

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Problem 7074

Differentiate $G(x)=-3 e^{x^{2}}$ without simplifying. Find $G^{\prime}(x)=$

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Problem 7075

Simplify $f(x+h)$ for $f(x)=-2x^2+5x$ and the difference quotient $\frac{f(x+h)-f(x)}{h}$ . Find $f'(x)$ .

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Problem 7076

Determine how fast the distance from the origin to the point on the curve $y=x^{2}+2$ changes if $\frac{d x}{d t}=2$ cm/s.

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Problem 7077

Find the derivative $\frac{d y}{d x}$ for $y=\frac{e^{x^{3}}}{\sqrt{8-x^{3}}$.

See Solution

Problem 7078

Differentiate implicitly: $(\sin \pi x + \cos \pi y)^{4} = 49$ . Find $\frac{dy}{dx}$ .

See Solution

Problem 7079

Identify the inner function $u=g(x)$ and outer function $y=f(u)$ for $y=\sqrt[3]{1+8x}$ . Find $\frac{dy}{dx}$ .

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Problem 7080

Find the derivative of $\sin(5x + 1)$ .

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Problem 7081

Find the derivative of $f(x)=5 e^{2 x^{10}-2 x^{8}}$ using the chain rule. $f^{\prime}(x)=$

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Problem 7082

Solve the equation $y^{\prime \prime}-2 y^{\prime}-3 y=4 x-5+6 x e^{2 x}$ using superposition and find the general solution.

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Problem 7083

Find the derivative $C'(100)$ of the cost function $C(x)=318+30x-0.05x^2+0.0006x^3$ and interpret it. Also, calculate the cost of producing the 101st item.

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Problem 7084

Find the derivative of $4^{5x+2}$ with respect to $x$ . No simplification needed.

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Problem 7085

Given values for $f(x), f^{\prime}(x), g(x), g^{\prime}(x)$ , find $(f g)^{\prime}(-9)$ and $\left(\frac{f}{g}\right)^{\prime}(-9)$ .

See Solution

Problem 7086

Calculate $\frac{d y}{d x}$ for $y=x^{x}$ where $x>0$ .

See Solution

Problem 7087

Find the derivative of the function $f(x)=\left(4 x^{2}+7 x+2\right) e^{-2 x}$ . What is $f^{\prime}(x)$ ?

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Problem 7088

Given functions $f(x)$ and $g(x)$ with their values, find $(f g)^{\prime}(-3)$ and $\left(\frac{f}{g}\right)^{\prime}(-3)$ .

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Problem 7089

Find the derivative of the function $f(x)=(4 x^{2}-5 x+2) \sin (5 x)$ . What is $f^{\prime}(x)$ ?

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Problem 7090

Calculez le changement d'enthalpie libre pour 2,3 mol d'un gaz idéal compressé isothermiquement de 1,5 atm à 6 atm à $298 \mathrm{~K}$ .

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Problem 7091

Integrate the equation: $\frac{d v}{d t}=\frac{50 t^{2}-80 \sqrt{t}}{5 \sqrt{t}}$ .

See Solution

Problem 7092

Find the velocity and acceleration of the spring mass at time $t=3$ for $s(t)=5 \sin(2t)$ . Round to the nearest hundredth.

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Problem 7093

A square's sides grow at $6 \mathrm{~cm/s}$ . Find the area growth rate when the area is $25 \mathrm{~cm}^2$ .

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Problem 7094

Find the derivative of $f(x)=\sin ^{2} x$ and evaluate $f^{\prime}(2)$ , rounding to the nearest hundredth.

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Problem 7095

A plane at 2 mi altitude flies at 480 mi/h. Find the distance increase rate when it's 4 mi from the radar station.

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Problem 7096

Analyze the beer market with demand $Q=140-4p$ and supply $Q=20+2p-42h$ . Find $\mathrm{dp}/\mathrm{dh}$ , $\mathrm{dQ}/\mathrm{dh}$ , and changes in $p$ and $Q$ when $h$ increases by \$2.

See Solution

Problem 7097

Differentiate $f(x)=\tan(2x)$ and simplify the result: $f^{\prime}(x)=$

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Problem 7098

At noon, ship A is 110 km west of ship B. A sails east at 25 km/h, B sails north at 15 km/h. Find the distance change rate at 4 PM.

See Solution

Problem 7099

Find the derivative $\frac{d y}{d x}$ for $y=\frac{e^{x^{4}}}{\sqrt{5-x^{4}}$.

See Solution

Problem 7100

Find $d y / d t$ for $y=7 \cos (x)$ at $x=\pi / 4$ .

See Solution

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Calculus

Problem 7001

Find dVdt\frac{d V}{d t}dtdV​ for a sphere where V=43πr3V=\frac{4}{3} \pi r^{3}V=34​πr3 in terms of drdt\frac{d r}{d t}dtdr​. Choose the correct option.

Problem 7002

Show how dGdt\frac{d G}{d t}dtdG​ relates to dMdt\frac{d M}{d t}dtdM​ by differentiating G=CM0.7G=\mathrm{CM}^{0.7}G=CM0.7 with respect to ttt. dGdt=□dMdt \frac{d G}{d t}=\square \frac{d M}{d t} dtdG​=□dtdM​

Problem 7003

A study shows tree growth rate GGG depends on body mass MMM as G=CM0.7G=CM^{0.7}G=CM0.7. Differentiate to find dGdt\frac{d G}{d t}dtdG​. dGdt=□dMdt \frac{d G}{d t}=\square \frac{d M}{d t} dtdG​=□dtdM​

Problem 7004

Differentiate G=CM0.7G=C M^{0.7}G=CM0.7 with respect to ttt and express dGdt\frac{d G}{d t}dtdG​ in terms of dMdt\frac{d M}{d t}dtdM​.

Problem 7005

Find dAdt\frac{d A}{d t}dtdA​ for a circle with area A=πr2A=\pi r^{2}A=πr2 in terms of drdt\frac{d r}{d t}dtdr​. Choose A, B, C, or D.

Problem 7006

A study shows tree growth rate G\mathrm{G}G depends on mass M\mathrm{M}M: G=CM0.7G=CM^{0.7}G=CM0.7. Find dGdt\frac{d G}{d t}dtdG​ and relate to dMdt\frac{d M}{d t}dtdM​.

Problem 7007

Rearrange the equation for fractional growth rates: dGdt=0.7CM0.7⋅1MdMdt \frac{d G}{d t}=0.7 \mathrm{CM}^{0.7} \cdot \frac{1}{M} \frac{d M}{d t} dtdG​=0.7CM0.7⋅M1​dtdM​ Show how 1GdGdt\frac{1}{G} \frac{d G}{d t}G1​dtdG​ relates to 1MdMdt\frac{1}{M} \frac{d M}{d t}M1​dtdM​.

Problem 7008

Find the slope of the tangent line to y=ln⁡(2cos⁡x)y=\ln (2 \cos x)y=ln(2cosx) at x=π4x=\frac{\pi}{4}x=4π​. Give your answer in simplest form.

Problem 7009

Finde die Ableitungsfunktion von f(x)=0,03x4−4x3+2xf(x)=0,03 x^{4}-4 x^{3}+2 xf(x)=0,03x4−4x3+2x.

Problem 7010

Find the integral of the function: ∫60x2e5x3dx\int 60 x^{2} e^{5 x^{3}} d x∫60x2e5x3dx

Problem 7011

Bestimme die Ableitungsfunktion von f(x)=0,03x4−4x3+2xf(x)=0,03 x^{4}-4 x^{3}+2 xf(x)=0,03x4−4x3+2x.

Problem 7012

Find the value of f′(3)f^{\prime}(3)f′(3) for the function f(x)=x2ln⁡(2x)f(x)=x^{2} \ln (2 x)f(x)=x2ln(2x).

Problem 7013

Bestimme die Ableitung von f(x)=6x5+5x3f(x)=6 x^{5}+5 x^{3}f(x)=6x5+5x3 mit Faktor- und Summenregel.

Problem 7014

Bestimme die Ableitung von f(x)=−3x5−2x+1f(x)=-3 x^{5}-2 x+1f(x)=−3x5−2x+1.

Problem 7015

Finde die Ableitung von f(x)=0,03x5−3x3+3xf(x)=0,03 x^{5}-3 x^{3}+3 xf(x)=0,03x5−3x3+3x.

Problem 7016

Find the limit: lim⁡x→8(3+x3)(2−6x2+x3)\lim _{x \rightarrow 8}(3+\sqrt[3]{x})(2-6 x^{2}+x^{3})x→8lim​(3+3x​)(2−6x2+x3). If none, enter DNE.

Problem 7017

Bestimme die Ableitung von f(x)=13x6+3x5+2f(x)=\frac{1}{3} x^{6}+3 x^{5}+2f(x)=31​x6+3x5+2.

Problem 7018

Solve the differential equation: xy′−2y=x2x y' - 2y = x^2xy′−2y=x2, for x>0x > 0x>0.

Problem 7019

Solve for yyy in the equation e5x3y=∫60x2e5x3dxe^{5 x^{3} y}=\int 60 x^{2} e^{5 x^{3}} d xe5x3y=∫60x2e5x3dx.

Problem 7020

Calculate the average velocity of a pebble from t=2t=2t=2 for (i) 0.1s, (ii) 0.05s, (iii) 0.01s using y=235−16t2y=235-16t^2y=235−16t2. Also, find instantaneous velocity at t=2t=2t=2.

Problem 7021

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=2x2+x−1f(x)=2x^{2}+x-1f(x)=2x2+x−1 and simplify your answer.

Problem 7022

Find the slope and equation of the tangent line to the curve 3xy−2πcos⁡(y)=5π3xy - 2\pi \cos(y) = 5\pi3xy−2πcos(y)=5π at the point (1,π)(1, \pi)(1,π).

Problem 7023

Problem 7024

Find the limit: lim⁡x→∞1+81x67−x3\lim _{x \rightarrow \infty} \frac{\sqrt{1+81 x^{6}}}{7-x^{3}}x→∞lim​7−x31+81x6​​. Enter ∞\infty∞, −∞-\infty−∞, or DNE if it doesn't exist.

Problem 7025

Berechne das Integral von f(x)=2−x3f(x)=2-\frac{x}{3}f(x)=2−3x​ über das Intervall [−4,6][-4, 6][−4,6].

Problem 7026

Find the derivative of SEC⁡x\operatorname{SEC} xSECx.

Problem 7027

Find the limit: lim⁡x→∞4x2−x+42x2+5x−7\lim _{x \rightarrow \infty} \frac{4 x^{2}-x+4}{2 x^{2}+5 x-7}x→∞lim​2x2+5x−74x2−x+4​.

Problem 7028

Find the xxx-values of critical points for y=x3(6−x)6y=x^{3}(6-x)^{6}y=x3(6−x)6 in the range -10 to 10.

Problem 7029

Find and simplify the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=2x2tan⁡xsec⁡xf(x)=\frac{2 x^{2} \tan x}{\sec x}f(x)=secx2x2tanx​.

Problem 7030

Find the tangent line equation for y=2tan⁡xy=2 \tan xy=2tanx at (π4,2)\left(\frac{\pi}{4}, 2\right)(4π​,2) in the form y=mx+by=m x+by=mx+b.

Problem 7031

Find the derivative of the function 2x2−csc⁡(x)+72 x^{2}-\csc (x)+72x2−csc(x)+7.

Problem 7032

Find the tangent line equation for y=4sec⁡x−8cos⁡xy=4 \sec x-8 \cos xy=4secx−8cosx at (π3,4)\left(\frac{\pi}{3}, 4\right)(3π​,4) in the form y=mx+by=m x+by=mx+b.

Problem 7033

Find the derivative of TAN⁡x\operatorname{TAN} xTANx.

Problem 7034

Find the derivative of f(x)=−2cos⁡x+5tan⁡xf(x)=-2 \cos x+5 \tan xf(x)=−2cosx+5tanx and evaluate f′(2π3)f^{\prime}\left(\frac{2 \pi}{3}\right)f′(32π​) rounded to the nearest hundredth.

Problem 7035

Problem 7036

Find the limit: lim⁡h→09+h−3h\lim _{h \rightarrow 0} \frac{\sqrt{9+h}-3}{h}h→0lim​h9+h​−3​. If it doesn't exist, write DNE.

Problem 7037

Find the derivative dydx\frac{d y}{d x}dxdy​ for y=8sec⁡(x)x5y=\frac{8 \sec (x)}{x^{5}}y=x58sec(x)​. No simplification needed. dydx=\frac{d y}{d x}=dxdy​=

Problem 7038

Find the limit: lim⁡x→−12x2+1x2+4x−5\lim _{x \rightarrow-1} \frac{2 x^{2}+1}{x^{2}+4 x-5}x→−1lim​x2+4x−52x2+1​. If it doesn't exist, write DNE.

Problem 7039

Differentiate the function using the Product Rule: f(x)=(6x+3)(x3−4)f(x)=(6x+3)(x^{3}-4)f(x)=(6x+3)(x3−4). Find f′(x)=f'(x)=f′(x)=.

Problem 7040

Differentiate the function using the Quotient Rule: f(x)=xx7+5f(x)=\frac{x}{x^{7}+5}f(x)=x7+5x​, find f′′(x)f^{\prime \prime}(x)f′′(x).

Problem 7041

Find $\frac{d V}{d t}$ for a sphere where $V=\frac{4}{3} \pi r^{3}$ in terms of $\frac{d r}{d t}$ . Choose the correct option.

Show how $\frac{d G}{d t}$ relates to $\frac{d M}{d t}$ by differentiating $G=\mathrm{CM}^{0.7}$ with respect to $t$ . $\frac{d G}{d t}=\square \frac{d M}{d t}$

A study shows tree growth rate $G$ depends on body mass $M$ as $G=CM^{0.7}$ . Differentiate to find $\frac{d G}{d t}$ . $\frac{d G}{d t}=\square \frac{d M}{d t}$

Differentiate $G=C M^{0.7}$ with respect to $t$ and express $\frac{d G}{d t}$ in terms of $\frac{d M}{d t}$ .

Find $\frac{d A}{d t}$ for a circle with area $A=\pi r^{2}$ in terms of $\frac{d r}{d t}$ . Choose A, B, C, or D.

A study shows tree growth rate $\mathrm{G}$ depends on mass $\mathrm{M}$ : $G=CM^{0.7}$ . Find $\frac{d G}{d t}$ and relate to $\frac{d M}{d t}$ .

Rearrange the equation for fractional growth rates:
$\frac{d G}{d t}=0.7 \mathrm{CM}^{0.7} \cdot \frac{1}{M} \frac{d M}{d t}$
Show how $\frac{1}{G} \frac{d G}{d t}$ relates to $\frac{1}{M} \frac{d M}{d t}$ .

Find the slope of the tangent line to $y=\ln (2 \cos x)$ at $x=\frac{\pi}{4}$ . Give your answer in simplest form.

Finde die Ableitungsfunktion von $f(x)=0,03 x^{4}-4 x^{3}+2 x$ .

Find the integral of the function: $\int 60 x^{2} e^{5 x^{3}} d x$

Bestimme die Ableitungsfunktion von $f(x)=0,03 x^{4}-4 x^{3}+2 x$ .

Find the value of $f^{\prime}(3)$ for the function $f(x)=x^{2} \ln (2 x)$ .

Bestimme die Ableitung von $f(x)=6 x^{5}+5 x^{3}$ mit Faktor- und Summenregel.

Bestimme die Ableitung von $f(x)=-3 x^{5}-2 x+1$ .

Finde die Ableitung von $f(x)=0,03 x^{5}-3 x^{3}+3 x$ .

Find the limit: $\lim _{x \rightarrow 8}(3+\sqrt[3]{x})(2-6 x^{2}+x^{3})$ . If none, enter DNE.

Bestimme die Ableitung von $f(x)=\frac{1}{3} x^{6}+3 x^{5}+2$ .

Solve the differential equation: $x y' - 2y = x^2$ , for $x > 0$ .

Solve for $y$ in the equation $e^{5 x^{3} y}=\int 60 x^{2} e^{5 x^{3}} d x$ .

Calculate the average velocity of a pebble from $t=2$ for (i) 0.1s, (ii) 0.05s, (iii) 0.01s using $y=235-16t^2$ . Also, find instantaneous velocity at $t=2$ .

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2x^{2}+x-1$ and simplify your answer.

Find the slope and equation of the tangent line to the curve $3xy - 2\pi \cos(y) = 5\pi$ at the point $(1, \pi)$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{1+81 x^{6}}}{7-x^{3}}$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

Berechne das Integral von $f(x)=2-\frac{x}{3}$ über das Intervall $[-4, 6]$ .

Find the derivative of $\operatorname{SEC} x$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{4 x^{2}-x+4}{2 x^{2}+5 x-7}$ .

Find the $x$ -values of critical points for $y=x^{3}(6-x)^{6}$ in the range -10 to 10.

Find and simplify the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{2 x^{2} \tan x}{\sec x}$ .

Find the tangent line equation for $y=2 \tan x$ at $\left(\frac{\pi}{4}, 2\right)$ in the form $y=m x+b$ .

Find the derivative of the function $2 x^{2}-\csc (x)+7$ .

Find the tangent line equation for $y=4 \sec x-8 \cos x$ at $\left(\frac{\pi}{3}, 4\right)$ in the form $y=m x+b$ .

Find the derivative of $\operatorname{TAN} x$ .

Find the derivative of $f(x)=-2 \cos x+5 \tan x$ and evaluate $f^{\prime}\left(\frac{2 \pi}{3}\right)$ rounded to the nearest hundredth.

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{9+h}-3}{h}$ . If it doesn't exist, write DNE.

Find the derivative $\frac{d y}{d x}$ for $y=\frac{8 \sec (x)}{x^{5}}$ . No simplification needed. $\frac{d y}{d x}=$

Find the limit: $\lim _{x \rightarrow-1} \frac{2 x^{2}+1}{x^{2}+4 x-5}$ . If it doesn't exist, write DNE.

Differentiate the function using the Product Rule: $f(x)=(6x+3)(x^{3}-4)$ . Find $f'(x)=$ .

Differentiate the function using the Quotient Rule: $f(x)=\frac{x}{x^{7}+5}$ , find $f^{\prime \prime}(x)$ .

Find the derivative $\frac{d y}{d x}$ for $y=\frac{8 \sec (x)}{x^{10}}$ . No need to simplify. $\frac{d y}{d x}=$

Find the derivative $f^{\prime}(x)$ and evaluate it at $c=0$ for $f(x)=(x^{4}+2x)(3x^{3}+5x-3)$ .

Calculate the integral $\int_{1}^{2} x^{5} \sqrt{x^{3}-1} \, dx$ .

Evaluate the integral $\int_{1}^{2} x^{5} \sqrt{x^{3}-1} \, dx$ .

Diana has 80 yards of fencing for a rectangle. Find the area $A$ as a function of width $W$ , max width for largest area, and max area.

Find the derivative $f^{\prime}(x)$ of $f(x)=\frac{x+5}{x-4}$ and calculate $f^{\prime}(9)$ .

Find the derivative of the function $f(x)=\frac{4-2x-x^{5}}{x^{3}-1}$ . What is $f^{\prime}(x)$ ?

Substitute $x=28$ into the rate of change expression: $-\frac{1}{79-28}$ . Round your answer to three decimal places.

Given $y^{2}+8x=x^{2}y-7$ and $y(4)=3$ , find $y'(4)$ and the tangent line equation at $(4,3)$ . $y=$

Find the volume change rate of a cylinder with radius $\sqrt{t+6}$ and height $\frac{1}{6} \sqrt{t}$ .

Find the acceleration of a car from rest at 5, 10, and 20 seconds using $v(t)=\frac{115 t}{8 t+15}$ . Round to three decimal places.

Find the second derivative of $f(x)=x^{4}+4x^{3}-7x^{2}-2x$ . What is $f^{\prime \prime}(x)$ ?

Find the second derivative of $f(x)=x \cos (x)$ , denoted as $f^{\prime \prime}(x)$ .

Find $f^{\prime}(2)$ given $g(2)=2$ , $g^{\prime}(2)=-3$ , $h(2)=-1$ , $h^{\prime}(2)=5$ , and $f(x)=\frac{g(x)}{h(x)}$ .

Find the second derivative of $f(x)=3x+37x^{-2}$ . What is $f^{\prime \prime}(x)$ ?

Find the derivative of the function $g(x)=\sqrt{x^{2}-2x+1}$ .

Find the derivative $f'(x)$ for $f(x)=(x^{2}+4x+5)^{3}$ and calculate $f'(4)$ .

Find $f^{\prime}(x)$ for $f(x)=\sqrt{4 x+6}$ and calculate $f^{\prime}(5)$ .

Find the slope of the tangent line to $y=\sin(3x)$ at the origin: $y'(0)=?$

Find the growth rate of infected people after 10 days using $I(t)=5500 e^{0.053 t}$ . Round to the nearest tenth.

Find the derivative of the function $f(x)=x^{5} e^{7.5 x}$ . What is $f^{\prime}(x)=?$

Find the growth rate of infected people after 11 days using $I(t)=3200 e^{0.035 t}$ . Round to the nearest tenth.

Find the slope of the tangent line to $y=\sin(3x)$ and $y=\sin(x)$ at the origin. Compare with cycles in $[0, 2\pi]$ .

If $h(x)=f(g(x))$ , find $h^{\prime}(1)$ using the values of $f$ and $g$ provided.

Find $F'(3)$ and $G'(3)$ given $F(x)=f(f(x))$ , $G(x)=(F(x))^2$ , and $f(3)=4$ , $f(4)=2$ , $f'(4)=15$ , $f'(3)=15$ .

Find the derivative of the function $f(y)=e^{-\sin y+6 \cos y}$ . What is $f^{\prime}(y)$ ?

Differentiate $f(w)=\cot(4w^3-9)$ and simplify the result. Find $f'(w)=$

Find the derivative of the function $f(x)=\frac{4}{(x^{2}-3x)^{2}}$ and evaluate it at $x=4$ .

Find the derivative $f'(4)$ of the function $f(x)=\frac{4}{(x^{2}-3x)^{2}}$ .

Find the rate of change of depth at 3 a.m. for the function $D(t)=4 \sin \left(\frac{\pi}{6} t+\frac{5 \pi}{6}\right)+2$ .

Find the derivative of $y=\frac{6}{x}+\sqrt{\cos (x)}$ at $(\pi / 2,12 / \pi)$ . Round to three decimal places.

Find the derivative of $-8 \sin^{2}(3x^{8})$ using the chain rule twice.

Differentiate $G(x)=-3 e^{x^{2}}$ without simplifying. Find $G^{\prime}(x)=$