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Calculus 2 : Calculus II

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

Example Questions

Example Question #283 : Derivatives

What is the derivative of the following function?

$f(x)=10^{sinx}$ .

Possible Answers:

$10^{cosx}*ln(10)$

$10^{sinx}*ln(10)*cos(x)$

$10^{sinx}*ln(10)*x$

$10^{sinx}*ln(sinx)$

$10^{sinx}*ln(10)$

Correct answer:

$10^{sinx}*ln(10)*cos(x)$

Explanation:

The derivative of a constant raised to a power is given by the following equation.

$\frac{d}{dx}a^u=a^ulna\frac{du}{dx}$

$\frac{d}{dx}10^{sinx}=10^{sinx}ln(10)\frac{d(sinx)}{dx}=10^{sinx}*ln(10)*cosx.$

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Example Question #1411 : Calculus Ii

Find the derivative of $f(x)=3x^{2/3}+8x^{-1/2}$

Possible Answers:

$f'(x)=2x^{-1/3}+8x^{1/2}$

$f'(x)=2x^{-1/3}-8x^{-3/2}$

$f'(x)=2x^{-1/3}+4x^{1/2}$

$f'(x)=2x^{-1/3}-4x^{-3/2}$

$f'(x)=2x^{-1/3}-4x^{1/2}$

Correct answer:

$f'(x)=2x^{-1/3}-4x^{-3/2}$

Explanation:

This derivative is an application of the power rule. Remember, we need to multiply the coefficient by the exponent, then reduce the exponent by one.

$f(x)=3x^{2/3}+8x^{-1/2}$

$f'(x)=3*\frac{2}{3}x^{2/3-1}+8*\frac{-1}{2}x^{-1/2-1}=2x^{-1/3}-4x^{-3/2}.$

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Example Question #1412 : Calculus Ii

Find the derivative of the function:

$f(x)=\cos(e^{5x})$

Possible Answers:

$-5e^{5x}\sin(e^{5x})$

$5e^{5x}\sin(e^{5x})$

$-e^{5x}\sin(e^{5x})$

$-5e^{4x}\sin(e^{5x})$

Correct answer:

$-5e^{5x}\sin(e^{5x})$

Explanation:

The derivative of the function is equal to

$f'(x)=-5e^{5x}\sin(e^{5x})$

and was found using the following rules:

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

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Example Question #289 : Derivatives

Find the derivative of the function:

$f(x)=\sec(x^2+x+10)$

Possible Answers:

$\sec(x^2+x+10)\tan(x^2+x+10)$

$(2x+1)\sec(x^2+x+10)\tan(x^2+x+10)$

$(2x+1)\ln\left |\sec(x^2+x+10)+\tan(x^2+x+10) \right |$

$-(2x+1)\sec(x^2+x+10)\tan(x^2+x+10)$

Correct answer:

$(2x+1)\sec(x^2+x+10)\tan(x^2+x+10)$

Explanation:

The derivative of the function is equal to

$f'(x)=(2x+1)\sec(x^2+x+10)\tan(x^2+x+10)$

and was found using the following rules:

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

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Example Question #1413 : Calculus Ii

Find the derivative of the function:

$f(x)=\frac{x}{e^x}$

Possible Answers:

$\frac{1-x}{e^x}$

$\frac{1}{e^x}$

$\frac{1-x}{e^{x^2}}$

$\frac{1+x}{e^x}$

Correct answer:

$\frac{1-x}{e^x}$

Explanation:

The derivative of the function is equal to

$f'(x)=\frac{1-x}{e^x}$

and was found using the following rules:

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

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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:

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

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

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

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

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

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 #291 : Derivative Review

Find the derivative of the function:

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

Possible Answers:

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

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

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

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

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 #91 : 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 #292 : Derivative Review

Find the derivative of the function:

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

Possible Answers:

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

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

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

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

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

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 #91 : First And Second Derivatives Of Functions

Find the derivative of the function:

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

Possible Answers:

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

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

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

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

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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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #283 : Derivatives

Example Question #1411 : Calculus Ii

Example Question #1412 : Calculus Ii

Example Question #289 : Derivatives

Example Question #1413 : Calculus Ii

Example Question #91 : First And Second Derivatives Of Functions

Example Question #291 : Derivative Review

Example Question #91 : First And Second Derivatives Of Functions

Example Question #292 : Derivative Review

Example Question #91 : First And Second Derivatives Of Functions

All Calculus 2 Resources

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