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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #2001 : Calculus

Find the first derivative of the following function:

$f(x)=\sec(e^{\frac{4}{x}})$

Possible Answers:

$-\frac{4}{x^2}e^{\frac{4}{x}}\sec(e^{\frac{4}{x}})\tan(e^{\frac{4}{x}})$

$-\frac{4}{x}e^{\frac{4}{x}}\sec(e^{\frac{4}{x}})\tan(e^{\frac{4}{x}})$

$\frac{4}{x^2}e^{\frac{4}{x}}\sec(e^{\frac{4}{x}})\tan(e^{\frac{4}{x}})$

$-\frac{4}{x^2}\sec(e^{\frac{4}{x}})\tan(e^{\frac{4}{x}})$

Correct answer:

$-\frac{4}{x^2}e^{\frac{4}{x}}\sec(e^{\frac{4}{x}})\tan(e^{\frac{4}{x}})$

Explanation:

The first derivative of the function is equal to

$f'(x)=-\frac{4}{x^2}e^{\frac{4}{x}}\sec(e^{\frac{4}{x}})\tan(e^{\frac{4}{x}})$

and was found using the following rules:

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

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Example Question #2001 : Calculus

Find the first derivative of the function:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first derivative of the function is equal to

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

and was found using the following rules:

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

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Example Question #2002 : Calculus

Find the derivative.

5-x^2

Possible Answers:

-2x

x^2

-x^2

Correct answer:

-2x

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}5=0$

$\frac{d}{dx}-x^2=-2x$

Thus, the derivative is -2x .

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Example Question #2003 : Calculus

Find the derivative.

$\frac{4}{x^3}$

Possible Answers:

$\frac{-8}{x^4}$

$\frac{8}{x^4}$

$\frac{12}{x^4}$

$\frac{-12}{x^4}$

Correct answer:

$\frac{-12}{x^4}$

Explanation:

Use the quotient rule to find the derivative. $\frac{x^3(0)-4(3x^2)}{(x^3)^2}=\frac{-12x^2}{x^6}=\frac{-12}{x^4}$

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Example Question #2004 : Calculus

Find the derivative.

5x^2(4x^3-2x)

Possible Answers:

100x^4-30x^2

70x^2

100x^4-20x^2

70x^4

Correct answer:

100x^4-30x^2

Explanation:

Use the product rule to find this derivative.

5x^2(12x^2-2)+(4x^3-2x)10x

Distribute.

60x^4-10x^2+40x^4-20x^2

Combine like terms.

100x^4-30x^2 is the derivative.

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Example Question #2005 : Calculus

Find the derivative at x=0.

4x^2-6x

Possible Answers:

Correct answer:

Explanation:

First, use the power rule to find the derivative.

$\frac{d}{dx}4x^2=8x$

$\frac{d}{dx}-6x=-6$

Thus, the derivative is 8x-6 .

Now, substitute 0 for x.

8(0)-6=-6

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Example Question #2006 : Calculus

Find the derivative at $x=\frac{1}{2}$ .

3x^2-6x

Possible Answers:

$3\frac{1}{2}$

Correct answer:

Explanation:

First, find the derivative using the power rule.

$\frac{d}{dx}3x^2=6x$

$\frac{d}{dx}-6x=-6$

Thus, the derivative is 6x-6 .

Now, substitute $\frac{1}{2}$ for x.

$6(\frac{1}{2})-6=-3$

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Example Question #2006 : Calculus

Find the derivative at x=5.

Possible Answers:

$\frac{4}{5}$

Correct answer:

Explanation:

First, find the derivative using the power rule.

$\frac{d}{dx}4x=4$ .

Thus, the derivative is 4 for all values of x.

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Example Question #2007 : Calculus

Find the derivative at x=2.

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

Possible Answers:

$\frac{-1}{2}$

$\frac{1}{2}$

$\frac{3}{4}$

$\frac{-3}{4}$

Correct answer:

$\frac{-3}{4}$

Explanation:

First, find the derivative using the quotient rule.

$\frac{x^2(0)-3(2x)}{(x^2)^2}=\frac{-6x}{x^4}=\frac{-6}{x^3}$

Thus, the derivative is $\frac{-6}{x^3}$ .

Now, substitute 2 for x.

$\frac{-6}{(2)^3}=\frac{-6}{8}=\frac{-3}{4}$

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Example Question #794 : How To Find Differential Functions

Find the first derivative of the following function at x=0:

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

Possible Answers:

$2\pi$

Correct answer:

Explanation:

The first derivative of the function is equal to

$f'(x)=(2x\sin(x)+x^2\cos(x))(e^{x^2\sin(x)})$

and was found using the following rules:

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

Finally, to evaluate the derivative at x=0, simply plug this into the first derivative function:

$f'(0)=(0+0)e^{0}=0$

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #2001 : Calculus

Example Question #2001 : Calculus

Example Question #2002 : Calculus

Example Question #2003 : Calculus

Example Question #2004 : Calculus

Example Question #2005 : Calculus

Example Question #2006 : Calculus

Example Question #2006 : Calculus

Example Question #2007 : Calculus

Example Question #794 : How To Find Differential Functions

All Calculus 1 Resources

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