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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

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9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #96 : First And Second Derivatives Of Functions

Find the derivative of the function:

$f(x)=\sqrt{\cos(x^2)}$

Possible Answers:

$\frac{{-x\sin(x^2)}{\sqrt{\cos(x^2)}}}{2}$

$\frac{-x\sin(x^2)}{2\sqrt{\cos(x^2)}}$

$\frac{-x\sin(x^2)}{\sqrt{\cos(x^2)}}$

$\frac{x\sin(x^2)}{\sqrt{\cos(x^2)}}$

Correct answer:

$\frac{-x\sin(x^2)}{\sqrt{\cos(x^2)}}$

Explanation:

The derivative of the function is equal to

$f'(x)=\frac{-x\sin(x^2)}{\sqrt{\cos(x^2)}}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f(g'(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$

Note that the square root itself is the "outer" function when using the first rule, the chain rule.

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Example Question #97 : First And Second Derivatives Of Functions

Find the derivative of the function:

$f(x)=\frac{x}{x+1}+3x\tan(x)$

Possible Answers:

$\frac{1}{(x+1)^2}+3\tan(x)+3x\sec^2(x)$

$\frac{2x+1}{(x+1)^2}+3\tan(x)+3x\sec^2(x)$

$\frac{1}{(x+1)^2}+3\tan(x)+3x\sec(x)$

$\frac{1}{(x+1)^2}+3x(\tan(x)+\sec^2(x))$

Correct answer:

$\frac{1}{(x+1)^2}+3\tan(x)+3x\sec^2(x)$

Explanation:

The derivative of the function is equal to

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

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \frac{f(x)}{g(x)}=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)$

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Example Question #98 : First And Second Derivatives Of Functions

Find the second derivative of the function:

$f(x)=e^{x^2}+\ln(x^2)$

Possible Answers:

$2xe^{x^2}+\frac{2}{x}$

$2xe^{x^2}+\left |\frac{2}{x} \right |$

$e^{x^2}+\frac{2}{x}$

$2xe^{x^2}+\frac{1}{x}$

Correct answer:

$2xe^{x^2}+\frac{2}{x}$

Explanation:

The derivative of the function is equal to

$f'(x)=2xe^{x^2}+\frac{2}{x}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{d} u}{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \ln(u)=\frac{1}{u}\frac{\mathrm{d} u}{\mathrm{d} x}$

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Example Question #1424 : Calculus Ii

Find the derivative of the function:

$f(x)=\sqrt[3]{x}+\sqrt{e^{x^2}}$

Possible Answers:

$\frac{1}{3\sqrt[3]{x^2}}+\frac{x\sqrt{e^{x^{2}}}}{2}$

$\frac{1}{3\sqrt[3]{x}}+x\sqrt{e^{x^{2}}}$

$\frac{1}{3\sqrt[3]{x^2}}+xe^x$

$\frac{1}{3\sqrt[3]{x^2}}+x\sqrt{e^{x^{2}}}$

Correct answer:

$\frac{1}{3\sqrt[3]{x^2}}+x\sqrt{e^{x^{2}}}$

Explanation:

The derivative of the function is equal to

$f'(x)=\frac{1}{3\sqrt[3]{x^2}}+x\sqrt{e^{x^{2}}}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{du} }{\mathrm{d} x}$

Note that all of the radicals act as "outer" functions when using the first rule, the chain rule.

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Example Question #99 : First And Second Derivatives Of Functions

Find the derivative of the function:

$f(x)=\cos(4x)\sec(2x)$

Possible Answers:

$-2\sin(2x)$

$2\sin(2x)$

$-4\sin(4x)\sec(2x)+\cos(4x)\sec(2x)\tan(2x)$

Correct answer:

$-4\sin(4x)\sec(2x)+\cos(4x)\sec(2x)\tan(2x)$

Explanation:

The derivative of the function is equal to

$f'(x)=-4\sin(4x)\sec(2x)+2\cos(4x)\sec(2x)\tan(2x)$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=\sin(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \sec(x)=\sec(x)\tan(x)$

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Example Question #301 : Derivative Review

Find the derivative of the function:

$f(x)=\sqrt[4]{xe^x}$

Possible Answers:

$\frac{e^x}{4\sqrt[4]{(xe^x)^3}}+\frac{\sqrt[4]{xe^x}}{4}$

$\frac{e^x}{4\sqrt[4]{(xe^x)^3}}+{\sqrt[4]{xe^x}}$

$\frac{e^x+1}{4\sqrt[4]{(xe^x)^3}}$

$\frac{e^x}{4\sqrt{(xe^x)^3}}+\frac{{\sqrt[4]{xe^x}}}{4}$

Correct answer:

$\frac{e^x}{4\sqrt[4]{(xe^x)^3}}+\frac{\sqrt[4]{xe^x}}{4}$

Explanation:

The derivative of the function is equal to

$f'(x)=\frac{e^x}{4\sqrt[4]{(xe^x)^3}}+\frac{\sqrt[4]{xe^x}}{4}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^x=e^x$

Note that the radical acts as the "outer" function using the first rule, the chain rule.

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Example Question #1421 : Calculus Ii

Find the derivative of the function:

f(x)=3^x+x^2

Possible Answers:

$\ln3+2x$

$x^3\ln3+2x$

$3^x\ln3+x^2$

$3^x\ln3+2x$

Correct answer:

$3^x\ln3+2x$

Explanation:

The derivative of the function is equal to

$f'(x)=3^x\ln3+2x$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} n^x=n^x \ln(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

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Example Question #303 : Derivative Review

Find the derivative of the function:

$f(x)=\ln(\sqrt{x+3}), x\geq-2$

Possible Answers:

$\frac{x+3}{2}$

$\frac{1}{2}$

$\frac{1}{2x+3}$

$\frac{1}{2x+6}$

Correct answer:

$\frac{1}{2x+6}$

Explanation:

The derivative of the function is equal to

$f'(x)=\frac{1}{2x+6}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \ln(u)=\frac{1}{u}\frac{\mathrm{du} }{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

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Example Question #304 : Derivative Review

Find the derivative of the function:

$f(x)=(\cos(x)+e^x)^4$

Possible Answers:

$4(\cos(x)+e^x)^3(\sin(x)+e^x)$

$4(\cos(x)+e^x)^3-\sin(x)+e^x$

$4(\cos(x)+e^x)^3$

$4(\cos(x)+e^x)^3(-\sin(x)+e^x)$

Correct answer:

$4(\cos(x)+e^x)^3(-\sin(x)+e^x)$

Explanation:

The derivative of the function is equal to

$f'(x)=4(\cos(x)+e^x)^3(-\sin(x)+e^x)$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x)=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^x=e^x$

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Example Question #305 : Derivative Review

Find the derivative of the function:

$f(x)=\csc(4x^2)$

Possible Answers:

$-\left |\csc(4x^2)+\cot(4x^2) \right |$

$-8x\csc(4x^2)\cot(4x^2)$

$8x\csc(4x^2)\cot(4x^2)$

$-\csc(4x^2)\cot(4x^2)$

Correct answer:

$-8x\csc(4x^2)\cot(4x^2)$

Explanation:

The derivative of the function is equal to

$f'(x)=-8x\csc(4x^2)\cot(4x^2)$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \csc(x)=-\csc(x)\cot(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #96 : First And Second Derivatives Of Functions

Example Question #97 : First And Second Derivatives Of Functions

Example Question #98 : First And Second Derivatives Of Functions

Example Question #1424 : Calculus Ii

Example Question #99 : First And Second Derivatives Of Functions

Example Question #301 : Derivative Review

Example Question #1421 : Calculus Ii

Example Question #303 : Derivative Review

Example Question #304 : Derivative Review

Example Question #305 : Derivative Review

All Calculus 2 Resources

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