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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #82 : Calculus Review

Find the derivative $\small f'(x)$ of the function $\small \small f(x)=\frac{1}{\sqrt{2x+1}}$

Possible Answers:

$\small \small \small \small \small \small f'(x)=-(2x+1)^{-\frac{3}{2}}$

$\small \small \small f'(x)=-\sqrt{2x+1}$

$\small \small \small \small \small \small \small f'(x)=-(2x+1)^{-\frac{1}{2}}$

$\small \small \small \small \small \small \small f'(x)=-\frac{1}{2}(2x+1)^{-\frac{3}{2}}$

Correct answer:

$\small \small \small \small \small \small f'(x)=-(2x+1)^{-\frac{3}{2}}$

Explanation:

We can find the derivative $\small f'(x)$ of the function $\small \small f(x)=\frac{1}{\sqrt{2x+1}}$ by using the power rule for derivatives:

$\frac{d}{dx}x^{a}=ax^{a-1}$

with $\small \small a=-\frac{1}{2}$ and the chain rule to get

$\small \small \small \small \small f'(x)=-\frac{1}{2}\cdot 2 (2x+1)^{-\frac{1}{2}-1}=-(2x+1)^{-\frac{3}{2}}$

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Example Question #83 : Calculus Review

Find the derivative $\small f'(x)$ of the function $\small \small \small f(x)=2x^{90}$

Possible Answers:

$\small \small \small \small \small \small \small \small f'(x)=\frac{1}{45}x^{91}$

$\small \small \small \small \small \small \small \small f'(x)=180x^{91}$

$\small \small \small \small \small \small \small f'(x)=180x^{89}$

$\small \small \small \small \small \small \small \small f'(x)=180x^{90}$

Correct answer:

$\small \small \small \small \small \small \small f'(x)=180x^{89}$

Explanation:

We can find the derivative $\small f'(x)$ of the function $\small \small \small f(x)=2x^{90}$ by using the power rule for derivatives:

$\frac{d}{dx}x^{a}=ax^{a-1}$

with $\small \small \small a=90$ and the chain rule to get

$\small \small \small \small \small \small f'(x)=2\cdot 90x^{89}=180x^{89}$

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Example Question #84 : Calculus Review

Find the derivative $\small f'(x)$ of the function $\small \small \small \small f(x)=5x^{95}$

Possible Answers:

$\small \small f'(x)=475x^{94}$

$\small \small \small f'(x)=\frac{5}{96}x^{96}$

$\small \small \small \small f'(x)=475x^{96}$

$\small \small \small f'(x)=475x^{95}$

Correct answer:

$\small \small f'(x)=475x^{94}$

Explanation:

We can find the derivative $\small f'(x)$ of the function $\small \small \small \small f(x)=5x^{95}$ by using the power rule for derivatives:

$\frac{d}{dx}x^{a}=ax^{a-1}$

with $\small \small \small \small a=95$ to get

$\small \small \small \small \small \small \small f'(x)=5\cdot 95x^{94}=475x^{94}$

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Example Question #85 : Calculus Review

Find the derivative $\small f'(x)$ of the function $\small \small \small \small \small f(x)=11x^{9}$

Possible Answers:

$\small \small \small \small \small \small \small \small \small f'(x)=99x^{8}$

$\small \small \small \small \small \small \small \small \small \small f'(x)=99x^{9}$

$\small \small \small \small f'(x)=\frac{11}{9}x^{10}$

$\small \small \small \small \small \small \small \small \small \small f'(x)=99x^{10}$

Correct answer:

$\small \small \small \small \small \small \small \small \small f'(x)=99x^{8}$

Explanation:

We can find the derivative $\small f'(x)$ of the function $\small \small \small \small \small f(x)=11x^{9}$ by using the power rule for derivatives:

$\frac{d}{dx}x^{a}=ax^{a-1}$

with $\small \small \small \small \small a=9$ to get

$\small \small \small \small \small \small \small \small f'(x)=11\cdot 9x^{8}=99x^{8}$

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Example Question #84 : Calculus Review

Find the derivative $\small f'(x)$ of the function $\small \small \small \small \small \small f(x)=4x^{2.5}$

Possible Answers:

$\small \small \small f'(x)=10x^{1.5}$

$\small \small \small \small f'(x)=10x^{2.5}$

$\small \small \small \small f'(x)=10\sqrt{x^5}$

$\small \small \small \small f'(x)=10\sqrt{x}$

Correct answer:

$\small \small \small f'(x)=10x^{1.5}$

Explanation:

We can find the derivative $\small f'(x)$ of the function $\small \small \small \small \small \small f(x)=4x^{2.5}$ by using the power rule for derivatives:

$\frac{d}{dx}x^{a}=ax^{a-1}$

with $\small \small \small \small \small \small a=2.5$ to get

$\small \small f'(x)=4\cdot 2.5x^{1.5}=10x^{1.5}$

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Example Question #81 : Calculus Review

Find the derivative $\small f'(x)$ of the function $\small \small f(x)=12x^{3}$

Possible Answers:

$\small \small \small \small f'(x)=36x^{2}$

$\small \small \small \small \small f'(x)=12x^{2}$

$\small \small \small \small \small f'(x)=3x^{2}$

$\small \small \small \small \small f'(x)=36x^{3}$

Correct answer:

$\small \small \small \small f'(x)=36x^{2}$

Explanation:

We can find the derivative $\small f'(x)$ of the function $\small \small f(x)=12x^{3}$ by using the power rule for derivatives:

$\frac{d}{dx}x^{a}=ax^{a-1}$

with $\small \small \small \small \small \small \small a=3$ to get

$\small \small \small f'(x)=12\cdot 3x^{2}=36x^{2}$

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Example Question #86 : Calculus Review

Find the derivative $\small f'(x)$ of the function $\small \small \small f(x)=\frac{\pi}{6}x^{10}$

Possible Answers:

$\small \small \small f'(x)=\frac{5\pi}{3}x^{10}$

$\small \small \small f'(x)=\frac{\pi}{66}x^{11}$

$\small \small \small f'(x)=\frac{10\pi}{3}x^{9}$

$\small \small f'(x)=\frac{5\pi}{3}x^{9}$

Correct answer:

$\small \small f'(x)=\frac{5\pi}{3}x^{9}$

Explanation:

We can find the derivative $\small f'(x)$ of the function $\small \small \small f(x)=\frac{\pi}{6}x^{10}$ by using the power rule for derivatives:

$\frac{d}{dx}x^{a}=ax^{a-1}$

with $\small \small a=10$ to get

$\small \small \small \small \small f'(x)=\frac{\pi}{6}\cdot 10x^{9}=\frac{5\pi}{3}x^{9}$

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Example Question #87 : Calculus Review

Find the derivative $\small f'(x)$ of the function $\small \small \small \small f(x)=(3x+1)^{2\pi}$

Possible Answers:

$\small \small \small f'(x)=6\pi(3x+1)^{2\pi+1}$

$\small \small \small f'(x)=2\pi(3x+1)^{2\pi-1}$

$\small \small \small \small f'(x)=3(3x+1)^{2\pi-1}$

$\small \small f'(x)=6\pi(3x+1)^{2\pi-1}$

Correct answer:

$\small \small f'(x)=6\pi(3x+1)^{2\pi-1}$

Explanation:

We can find the derivative $\small f'(x)$ of the function $\small \small \small \small f(x)=(3x+1)^{2\pi}$ by using the power rule for derivatives:

$\frac{d}{dx}x^{a}=ax^{a-1}$

with $\small \small \small a=2\pi$ and the chain rule to get

$\small \small \small \small \small \small \small f'(x)=3\cdot 2\pi(3x+1)^{2\pi-1}=6\pi(3x+1)^{2\pi-1}$

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Example Question #88 : Calculus Review

Find the derivative $\small f'(x)$ of the function $\small \small \small \small \small f(x)=(x^3+1)^{e^e}$

Possible Answers:

$\small \small f'(x)=3x^2\cdot e^e(x^3+1)^{e^e-1}$

$\small \small \small f'(x)=x^2\cdot e^e(x^3+1)^{e^e-1}$

$\small \small \small \small f'(x)=x^3\cdot e^e(x^3+1)^{e^e}$

$\small \small \small f'(x)=3x^2\cdot e^e(x^3+1)^{e^e+1}$

Correct answer:

$\small \small f'(x)=3x^2\cdot e^e(x^3+1)^{e^e-1}$

Explanation:

We can find the derivative $\small f'(x)$ of the function $\small \small \small \small \small f(x)=(x^3+1)^{e^e}$ by using the power rule for derivatives:

$\frac{d}{dx}x^{a}=ax^{a-1}$

with $\small \small \small \small a=e^e$ and the chain rule to get

$\small \small f'(x)=3x^2\cdot e^e(x^3+1)^{e^e-1}$

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Example Question #91 : Derivatives

Find the derivative $\small f'(x)$ of the function $\small \small \small \small \small \small \small f(x)=(2^{95}e^\pi+10!)x$

Possible Answers:

$\small \small \small \small f'(x)=\frac{2^{95}e^\pi+10!}{2}x^2$

$\small \small \small \small f'(x)=(2^{95}e^\pi+10!)x^2$

$\small \small \small \small f'(x)=2^{95}e^\pi+10!-1$

$\small \small \small f'(x)=2^{95}e^\pi+10!$

Correct answer:

$\small \small \small f'(x)=2^{95}e^\pi+10!$

Explanation:

We can find the derivative $\small f'(x)$ of the function $\small \small \small \small \small \small \small f(x)=(2^{95}e^\pi+10!)x$ by using the power rule for derivatives:

$\frac{d}{dx}x^{a}=ax^{a-1}$

with $\small \small \small \small \small a=1$ to get

$\small \small f'(x)=(2^{95}e^\pi+10!)\cdot 1x^0=2^{95}e^\pi+10!$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #82 : Calculus Review

Example Question #83 : Calculus Review

Example Question #84 : Calculus Review

Example Question #85 : Calculus Review

Example Question #84 : Calculus Review

Example Question #81 : Calculus Review

Example Question #86 : Calculus Review

Example Question #87 : Calculus Review

Example Question #88 : Calculus Review

Example Question #91 : Derivatives

All Calculus 3 Resources

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