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Calculus 2 : First and Second Derivatives of Functions

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Example Questions

Example Question #91 : First And Second Derivatives Of Functions

Find the derivative of the function:

$f(x)=\frac{1}{x}+x^2+5x+e^{\tan(x)}$

Possible Answers:

$-\frac{1}{x^2}+2x+5+e^{\tan(x)}$

$-\frac{1}{x^2}+7x+\sec^2(x)(e^{\tan(x)})$

$-\frac{1}{x^2}+2x+5+\sec(x)(e^{\tan(x)})$

$\frac{1}{x^2}+2x+5+\sec^2(x)(e^{\tan(x)})$

$-\frac{1}{x^2}+2x+5+\sec^2(x)(e^{\tan(x)})$

Correct answer:

$-\frac{1}{x^2}+2x+5+\sec^2(x)(e^{\tan(x)})$

Explanation:

The derivative of the function is equal to

$f'(x)=-\frac{1}{x^2}+2x+5+\sec^2(x)(e^{\tan(x)})$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{d} u}{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)$

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Example Question #92 : First And Second Derivatives Of Functions

Find the derivative of the function:

$f(x)=\tan(x^2)+x^3$

Possible Answers:

$\sec^2(x^2)+3x^2$

$2x\sec(x^2)+3x^2$

$2x\sec^2(x^2)+3x^2$

$\sec^2(x^2)\tan(x^2)+3x^2$

Correct answer:

$2x\sec^2(x^2)+3x^2$

Explanation:

The derivative of the function is equal to

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

and was found using the following rules:

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

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Example Question #93 : First And Second Derivatives Of Functions

Find the derivative of the function:

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

Possible Answers:

$2x\sin(2x^4)+8x^6\cos(2x^4)$

$2x\sin(2x^4)-8x^5\cos(2x^4)$

$2x\sin(2x^4)+8x^5\cos(2x^4)$

$x^2\sin(2x^4)+8x^5\cos(2x^4)$

Correct answer:

$2x\sin(2x^4)+8x^5\cos(2x^4)$

Explanation:

The derivative of the function is equal to

$f'(x)=2x\sin(2x^4)+8x^5\cos(2x^4)$

and was found using the following rules:

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

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Example Question #94 : First And Second Derivatives Of Functions

Find the derivative of the function:

$f(x)=e^{3x^2}\cos(3x)$

Possible Answers:

$6xe^{3x^2}\cos(3x)-3e^{3x^2}\sin(3x)$

$6x\cos(3x)-3e^{3x^2}\sin(3x)$

$6xe^{3x^2}\cos(3x)-e^{3x^2}\sin(3x)$

$6xe^{3x^2}\cos(3x)+3e^{3x^2}\sin(3x)$

$3x^2e^{3x^2}\cos(3x)-3e^{3x^2}\sin(3x)$

Correct answer:

$6xe^{3x^2}\cos(3x)-3e^{3x^2}\sin(3x)$

Explanation:

The derivative of the function is equal to

$f'(x)=6xe^{3x^2}\cos(3x)-3e^{3x^2}\sin(3x)$

and was found using the following rules:

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

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Example Question #95 : First And Second Derivatives Of Functions

Find the derivative of the function:

$f(x)=x^2+xe^{2x}+10x$

Possible Answers:

$e^{2x}(2x+1)+12x$

$e^{2x}(2x+1)+2x+10$

$2xe^{2x}+2x+10$

$e^{2x}(x+1)+2x+10$

Correct answer:

$e^{2x}(2x+1)+2x+10$

Explanation:

The derivative of the function is equal to

$f'(x)=e^{2x}(2x+1)+2x+10$

and was found using the following rules:

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

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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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Calculus 2 : First and Second Derivatives of Functions

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #91 : First And Second Derivatives Of Functions

Example Question #92 : First And Second Derivatives Of Functions

Example Question #93 : First And Second Derivatives Of Functions

Example Question #94 : First And Second Derivatives Of Functions

Example Question #95 : First And Second Derivatives Of Functions

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

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