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Calculus 3 : Derivatives

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Example Questions

Example Question #101 : Calculus Review

Find the derivative of the function $h(x)=\sec x\tan x$

Possible Answers:

$h'(x)=\sec x\tan^2x+\sec^2x$

$h'(x)=\sec x\tan^3x+\sec^3x$

$h'(x)=\sec^3 x\tan x$

$h'(x)=\sec x\tan^2x+\sec^3x$

Correct answer:

$h'(x)=\sec x\tan^2x+\sec^3x$

Explanation:

To find the derivative of the function $h(x)=\sec x\tan x$ , we use the product rule:

[f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)

with $f(x)=\sec x$ and $g(x)=\tan x$

so we get

$h'(x)=\sec x \tan x\cdot \tan x+\sec x\cdot \sec^2 x=\sec x\tan^2x+\sec^3x$

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Example Question #102 : Calculus Review

Find the derivative of the function $h(x)=\csc x\tan x$

Possible Answers:

$h'(x)=-\csc x \cot x\cdot \tan x+\csc x\cdot \sec x$

$h'(x)=-\csc x \cot x\cdot \tan x+\csc x\cdot \sec^2 x$

$h'(x)=\csc x \cot x\cdot \tan x-\csc x\cdot \sec^2 x$

$h'(x)=\csc x \cot x\cdot \tan x+\csc x\cdot \sec^2 x$

Correct answer:

$h'(x)=-\csc x \cot x\cdot \tan x+\csc x\cdot \sec^2 x$

Explanation:

To find the derivative of the function $h(x)=\csc x\tan x$ , we use the product rule:

[f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)

with $f(x)=\csc x$ and $g(x)=\tan x$

so we get

$h'(x)=-\csc x \cot x\cdot \tan x+\csc x\cdot \sec^2 x$

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Example Question #103 : Calculus Review

Find the derivative of the function $h(x)=e^{x}\cos x$

Possible Answers:

$h'(x)=-e^x(\cos x+\sin x)$

$h'(x)=e^x(\cos x-\sin x)$

$h'(x)=e^x(-\cos x+\sin x)$

$h'(x)=e^x(\cos x+\sin x)$

Correct answer:

$h'(x)=e^x(\cos x-\sin x)$

Explanation:

To find the derivative of the function $h(x)=e^{x}\cos x$ , we use the product rule:

[f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)

with f(x)=e^x and $g(x)=\cos x$

so we get

$h'(x)=e^x\cdot \cos x+e^x\cdot -\sin x=e^x(\cos x-\sin x)$

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Example Question #104 : Calculus Review

Find the derivative of the function $h(x)=e^{\pi x}\sec x$

Possible Answers:

$h'(x)=e^{\pi x}(\sec x+\sec^2 x\tan x)$

$h'(x)=e^{\pi x}(\sec^2 x+\sec x\tan x)$

$h'(x)=e^{\pi x}(\pi\sec x+\tan x)$

$h'(x)=e^{\pi x}(\pi\sec x+\sec x\tan x)$

Correct answer:

$h'(x)=e^{\pi x}(\pi\sec x+\sec x\tan x)$

Explanation:

To find the derivative of the function $h(x)=e^{\pi x}\sec x$ , we use the product rule:

[f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)

with $f(x)=e^{\pi x}$ and $g(x)=\sec x$

so we get

$h'(x)=\pi e^{\pi x}\cdot \sec x+e^{\pi x}\cdot \sec x\tan x=e^{\pi x}(\pi\sec x+\sec x\tan x)$

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Example Question #105 : Calculus Review

Find the derivative of the function $h(x)=(2+x^3)\sec x$

Possible Answers:

$h'(x)=3x^2\cdot \sec x+(2+x^3)\cdot \sec x\tan x$

$h'(x)=3x^2\cdot \sec x+(3x^2)\cdot \sec x\tan x$

$h'(x)=3x^2\cdot \sec x+(2+x^3)\cdot \tan x$

$h'(x)=3x^2\cdot \sec^2 x+(2+x^3)\cdot \sec x\tan x$

Correct answer:

$h'(x)=3x^2\cdot \sec x+(2+x^3)\cdot \sec x\tan x$

Explanation:

To find the derivative of the function $h(x)=(2+x^3)\sec x$ , we use the product rule:

[f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)

with f(x)=2+x^3 and $g(x)=\sec x$

so we get

$h'(x)=3x^2\cdot \sec x+(2+x^3)\cdot \sec x\tan x$

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Example Question #106 : Calculus Review

Find the derivative of the function $h(x)=x^5\ln x$

Possible Answers:

$h'(x)=5x^4\ln x+5x^5$

$h'(x)=5x^4\ln x+5x^4$

$h'(x)=5x^4\ln x+x^4$

$h'(x)=5x^4\ln x+x^5$

Correct answer:

$h'(x)=5x^4\ln x+x^4$

Explanation:

To find the derivative of the function $h(x)=x^5\ln x$ , we use the product rule:

[f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)

with f(x)=x^5 and $g(x)=\ln x$

so we get

$h'(x)=5x^4\cdot \ln x+x^5\cdot \frac{1}{x}=5x^4\ln x+x^4$

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Example Question #107 : Calculus Review

Find the derivative of the function $h(x)=x^2\ln x$

Possible Answers:

h'(x)=2x

$h'(x)=x(2\ln x+2)$

$h'(x)=x(2\ln x+1)$

h'(x)=2

Correct answer:

$h'(x)=x(2\ln x+1)$

Explanation:

To find the derivative of the function $h(x)=x^2\ln x$ , we use the product rule:

[f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)

with f(x)=x^2 and $g(x)=\ln x$

so we get

$h'(x)=2x\cdot \ln x+x^2\cdot \frac{1}{x}=2x\ln x+x=x(2\ln x+1)$

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Calculus 3 : Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #101 : Calculus Review

Example Question #102 : Calculus Review

Example Question #103 : Calculus Review

Example Question #104 : Calculus Review

Example Question #105 : Calculus Review

Example Question #106 : Calculus Review

Example Question #107 : Calculus Review

All Calculus 3 Resources

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