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AP Calculus AB : Derivatives

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Example Questions

Example Question #61 : Derivatives

Given the function $\small \small \small \small \small z(x)=x^5-x^6-100$ , find its derivative.

Possible Answers:

$\small z'(x)=5x^4-6x^5$

$\small \small z'(x)=5x^4-6x^5-100$

$\small \small z'(x)=-5x^4+6x^5$

$\small \small z'(x)=5x^4-6x^5+100$

Correct answer:

$\small z'(x)=5x^4-6x^5$

Explanation:

Given the function $\small \small \small \small \small z(x)=x^5-x^6-100$ , we can find its derivative using the power rule, which states that

$\small \small [x^n]'=nx^{n-1}$

So we have

$\small z'(x)=5x^4-6x^5$

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Example Question #62 : Derivatives

Given the function $\small \small \small \small \small \small z(x)=x^\pi-x^{100}$ , find its derivative.

Possible Answers:

$\small \small \small z'(x)=\pi x^{\pi}-100x^{99}$

$\small \small z'(x)=\pi x^{\pi-1}-100x^{99}$

$\small \small \small z'(x)=\pi x^{\pi+1}-100x^{99}$

$\small \small \small z'(x)=\pi x^{\pi-1}-x^{100}$

Correct answer:

$\small \small z'(x)=\pi x^{\pi-1}-100x^{99}$

Explanation:

Given the function $\small \small \small \small \small \small z(x)=x^\pi-x^{100}$ , we can find its derivative using the power rule, which states that

$\small \small [x^n]'=nx^{n-1}$

So we have

$\small \small z'(x)=\pi x^{\pi-1}-100x^{99}$

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Example Question #63 : Derivatives

Given the function $\small \small \small \small \small \small \small z(x)=ex^5+\pi x^2$ , find its derivative.

Possible Answers:

$\small \small \small \small z'(x)=5ex^4+2\pi$

$\small \small \small \small z'(x)=5ex^5+2\pi x$

$\small \small \small \small z'(x)=5e^{5x}+2\pi x$

$\small \small \small z'(x)=5ex^4+2\pi x$

Correct answer:

$\small \small \small z'(x)=5ex^4+2\pi x$

Explanation:

Given the function $\small \small \small \small \small \small \small z(x)=ex^5+\pi x^2$ , we can find its derivative using the power rule, which states that

$\small \small [x^n]'=nx^{n-1}$

So we have

$\small \small \small z'(x)=5ex^4+2\pi x$

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Example Question #64 : Derivatives

Find the first derivative of the function:

$f(x)=\cos(x^2)+\tan(x^3)+x^4$

Possible Answers:

$-\sin(x^2)+\sec^2(x^3)+4x^3$

$2x\sin(x^2)+3x^2\sec^2(x^3)+4x^3$

$-2x\sin(x^2)+3x^2\sec^2(x^3)+4x^3$

$-2x\sin(x^2)+3x^2\sec^2(x^3)+4x$

Correct answer:

$-2x\sin(x^2)+3x^2\sec^2(x^3)+4x^3$

Explanation:

The derivative of the function is equal to

$f'=-2x\sin(x^2)+3x^2\sec^2(x^3)+4x^3$

and was found using the following rules:

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

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Example Question #61 : Derivatives

Find the derivative.

$3x^3-2x^\frac{1}{2}$

Possible Answers:

6x^2-x

$9x^2-\frac{1}{x^\frac{1}{2}}$

$6x^2-\frac{1}{x^\frac{1}{2}}$

9x^2-x

Correct answer:

$9x^2-\frac{1}{x^\frac{1}{2}}$

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}3x^3=9x^2$

$\frac{d}{dx}-2x^\frac{1}{2}=-\frac{1}{x^\frac{1}{2}}$

Thus, the derivative is $9x^2-\frac{1}{x^\frac{1}{2}}$ .

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Example Question #62 : Derivatives

Find the derivative.

x^4-3x+24

Possible Answers:

4x-3

4x^3-3

x^3-x

4x^3+18

Correct answer:

4x^3-3

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}x^4=4x^3$

$\frac{d}{dx}-3x=-3$

$\frac{d}{dx}24=0$

Thus, the derivative is 4x^3-3 .

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Example Question #51 : Derivatives Of Functions

Find the derivative.

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

Possible Answers:

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

$\frac{1}{2x}$

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

Correct answer:

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

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}x^\frac{1}{2}=\frac{1}{2x^\frac{1}{2}}$

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Example Question #52 : Derivatives Of Functions

Find the derivative.

$\sin (x)$

Possible Answers:

$\cos (x)$

$-\sin (x)$

$-\cos (x)$

Correct answer:

$\cos (x)$

Explanation:

The derivative of $\sin (x)$ is $\cos (x)$ . (Memorization)

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Example Question #53 : Derivatives Of Functions

Find the derivative.

$\sin (2x)$

Possible Answers:

$2\cos (2x)$

$\cos (2)$

$\sin (2)$

$-2\sin (2x)$

Correct answer:

$2\cos (2x)$

Explanation:

Use the chain rule to find the derivative: $\frac{d}{dx}f(g(x))=g'(x)f'(g(x))$

Thus, $\frac{d}{dx}\sin (2x)=2\cos (2x)$ .

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Example Question #54 : Derivatives Of Functions

Find the derivative.

7x^2+4x-2

Possible Answers:

7x-4

7x+4

14x+4

14x-4

Correct answer:

14x+4

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}7x^2=14x$

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

The derivative of a constant is zero.

Thus, the derivative is 14x+4 .

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AP Calculus AB : Derivatives

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #61 : Derivatives

Example Question #62 : Derivatives

Example Question #63 : Derivatives

Example Question #64 : Derivatives

Example Question #61 : Derivatives

Example Question #62 : Derivatives

Example Question #51 : Derivatives Of Functions

Example Question #52 : Derivatives Of Functions

Example Question #53 : Derivatives Of Functions

Example Question #54 : Derivatives Of Functions

All AP Calculus AB Resources

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