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AP Calculus AB : Derivatives of functions

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Example Questions

Example Question #61 : Derivatives Of Functions

Given $f(x)=2x^{3}-7x^{2}+8x-3$ . Find f''(2) .

Possible Answers:

Correct answer:

Explanation:

First, find the first derivative.

You should get $f'(x)=6x^{2}-14x+8$ .

Next, differentiate again.

You should get f''(x)=12x-14 .

Finally, plug in x=2 to get f''(2)=10 .

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Example Question #62 : Derivatives Of Functions

The velocity profile of a fluid is given by

$v(x)=16\cos(x)+x^2+16\sin(x)$

Determine the rate of change of the velocity of the fluid at any point.

Possible Answers:

$16(\cos(x)-\sin(x))+x^2$

$16(\cos(x)-\sin(x))+2x$

$16(\cos(x)+\sin(x))+2x$

2x+16

Correct answer:

$16(\cos(x)-\sin(x))+2x$

Explanation:

To find the rate of change of the velocity of the fluid at any point, we must take the derivative of the function:

$v'(x)=-16\sin(x)+2x+16\cos(x)=16(\cos(x)-\sin(x))+2x$

The derivative was found using the following rules:

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

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Example Question #61 : Derivatives Of Functions

Find :

$f=\sqrt{\tan(4x)}$

Possible Answers:

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

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

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

$\frac{1}{\sqrt{\tan(4x)}}$

Correct answer:

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

Explanation:

The derivative of the function is equal to

$f'=\frac{1}{2}\tan(4x)^{-\frac{1}{2}}(4\sec^2(4x))=\frac{2\sec^2(4x)}{\sqrt{\tan(4x)}}$

and was found using the following rules:

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

Note that the chain rule was used on the square root, the inner tangent function, and the function inside the secant.

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Example Question #61 : Derivatives Of Functions

Find the derivative of the function:

$f=\cos^2(2x)$

Possible Answers:

$4\cos(2x)\sin(2x)$

$-4\cos(2x)\sin(2x)$

$-4\cos(4x)\sin(4x)$

$-2\cos(2x)\sin(2x)$

Correct answer:

$-4\cos(2x)\sin(2x)$

Explanation:

The derivative of the function is equal to

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

and was found using the following rules:

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

Notice that the chain rule was used on the cosine squared function, the cosine function, and the function inside the sine function.

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Example Question #61 : Derivatives Of Functions

Find the derivative of the function

$f=x\cos(x^2)$

Possible Answers:

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

$f'=\cos(x^2)-x\sin(x^2)$

$f'=-\cos(x^2)-2x^2\sin(x^2)$

$f'=\cos(x^2)-2x^2\sin(x^2)$

Correct answer:

$f'=\cos(x^2)-2x^2\sin(x^2)$

Explanation:

To find the derivative of the function, we use both the product rule and the chain rule

$f'=x'\cos(x^2)+x\cos(x^2)'=\cos(x^2)-x\sin(x^2)(2x)=\cos(x^2)-2x^2\sin(x^2)$

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Example Question #62 : Derivatives Of Functions

Find the derivative of the function $f=\frac{x^5}{\sin(2x)}$

Possible Answers:

$f'=\frac{5x^4\sin(2x)-2x^5\cos(2x)}{\cos^2(2x)}$

$f'=\frac{5x^4\sin(2x)-2x^5\cos(2x)}{\sin^2(2x)}$

$f'=\frac{5x^4\sin(2x)+2x^5\cos(2x)}{\sin^2(2x)}$

$f'=\frac{5x^4\sin(x)-2x^5\cos(2x)}{\sin^2(2x)}$

Correct answer:

$f'=\frac{5x^4\sin(2x)-2x^5\cos(2x)}{\sin^2(2x)}$

Explanation:

To find the derivative of the function, we use both the quotient rule and the chain rule

$f'=(x^5'*\sin(2x)-x^5\sin(2x)')/\sin^2(2x))$

$f'=\frac{5x^4\sin(2x)-2x^5\cos(2x)}{\sin^2(2x)}$

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Example Question #63 : Derivatives Of Functions

If $f(x)=\sin x$ then $f'\left({\frac{\pi }{4}}\right)=$

Possible Answers:

$\frac{{\sqrt{3}}}{2}$

$\frac{\sqrt{2}}{2}$

Correct answer:

$\frac{\sqrt{2}}{2}$

Explanation:

To calculate the derivative of this function at the desired point, first recall that,

$\sin(x)\frac{d}{dx}=\cos(x)$

$f(x)=\sin x$

$f^{'}(x)=\cos x$

Now, substitute the value into the derivative function to solve.

$f'(\frac{\pi }{4}) = cos\frac{\pi }{4} = cos(45^{\circ}) = \frac{\sqrt{2}}{2}$

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Example Question #64 : Derivatives Of Functions

Let $f(x)=x^2-\frac{1}{1-x^2}$ . Which of the following gives the equation of the line normal to $f(x)$ when x=2 ?

Possible Answers:

$27x+111y=535$

$27x + 96y = 470$

$32x+9y=103$

$192x + 54y = 735$

$32x-9y=25$

Correct answer:

$27x + 96y = 470$

Explanation:

We are asked to find the normal line. This means we need to find the line that is perpendicular to the tangent line at x=2 . In order to find the tangent line, we will need to evaluate the derivative of f(x) at x=2 .

$f(x)=x^2-\frac{1}{1-x^2}=x^2-(1-x^2)^{-1}$

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

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

$f'(2)=2(2)-2(2)(1-2^2)^{-2}$

$f'(2)=4-4(\frac{1}{9})=\frac{32}{9}$

The slope of the tangent line at x=2 is $\frac{32}{9}$ . Because the tangent line and the normal line are perpendicular, the product of their slopes must equal .

(slope of tangent)(slope of normal) =

$\left ( \frac{32}{9} \right )\cdot \left ( slope\ of\ normal \right )=-1$

$slope\ of\ normal=-\frac{9}{32}$

We now have the slope of the normal line. Once we find a point through which it passes, we will have enough information to derive its equation.

Since the normal line passes through the function at x=2 , it will pass through the point $\left ( 2,f(2) \right )$ . Be careful to use the original equation for , not its derivative.

$f(2)=2^2-(1-4)^{-1}=4-(-\frac{1}{3})=\frac{13}{3}$

The normal line has a slope of $-\frac{9}{32}$ and passes through the piont $\left ( 2,\frac{13}{3} \right )$ . We can now use point-slope form to find the equation of the normal line.

$y-\frac{13}{3}=-\frac{9}{32}(x-2)$

Multiply both sides by 32(3)=96 .

$96y-416=-27(x-2)$

$27x + 96y = 470$

The answer is $27x + 96y = 470$ .

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Example Question #65 : Derivatives Of Functions

If the position of a particle over time is represented by $p=t^{3}-16t^{2}-7t$ then what is the particle's instantaneous acceleration at t=4 ?

Possible Answers:

Correct answer:

Explanation:

The answer is .

Since velocity is the first derivative of the position function, take the derivative once. Then, recall that the acceleration function is the second derivative of position thus the derivative needs to be taken one more time.

$p=t^{3}-16t^{2}-7t$

$velocity =p'=3t^{2}-32t-7$

$acceleration =p''=6t-32$

$p''(4)=6(4)-32 = -8$

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Example Question #66 : Derivatives Of Functions

Consider the function:

$f(x) = -6e^{-x} + x$

The relative minimum for this function is at:

Possible Answers:

There is no relative minimum, because the first derivative is always positive.

$x = \frac{-1}{6}$

None of the other answers.

x = 0

x = 10

Correct answer:

There is no relative minimum, because the first derivative is always positive.

Explanation:

To find any relative minimum, one first needs to find the critical points by setting the first derivative equal to zero:

$f'(x) = 6e^{-x} + 1 = 0$

However, the first derivative is positive for all real values of x, since the exponential function is always positive. Thus, there are no values for which f'(x) = 0 , and therefore no critical points and no relative minimum.

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AP Calculus AB : Derivatives of functions

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #61 : Derivatives Of Functions

Example Question #62 : Derivatives Of Functions

Example Question #61 : Derivatives Of Functions

Example Question #61 : Derivatives Of Functions

Example Question #61 : Derivatives Of Functions

Example Question #62 : Derivatives Of Functions

Example Question #63 : Derivatives Of Functions

Example Question #64 : Derivatives Of Functions

Example Question #65 : Derivatives Of Functions

Example Question #66 : Derivatives Of Functions

All AP Calculus AB Resources

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