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AP Calculus AB : Computation of the Derivative

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Example Questions

Example Question #58 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following function using the quotient rule:

$f=\frac{e^x}{\tan{x}}$

Possible Answers:

$f'=\frac{e^x\tan{x}+e^x\sec^2{x}}{\tan^2{x}}$

$f'=\frac{xe^x\tan{x}-e^x\sec^2{x}}{\tan^2{x}}$

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

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

Correct answer:

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

Explanation:

To find the derivative of the function using the quotient rule, we apply the following definition:

$\frac{\mathrm{d} }{\mathrm{d} x}(\frac{f}{g})=\frac{gf'-fg'}{g^2}$

$\frac{\mathrm{d} }{\mathrm{d} x}(\frac{e^x}{\tan{x}})=\frac{\tan{x}*\frac{\mathrm{d} }{\mathrm{d} x}(e^x)-e^x*\frac{\mathrm{d} }{\mathrm{d} x}(\tan{x})}{\tan^2{x}}=\frac{e^x\tan{x}-e^x\sec^2{x}}{\tan^2{x}}$

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Example Question #52 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following function:

f=10x^3+5x^2-1

Possible Answers:

f'=30x^2+10x

f'=3x^2+x

f'=30x^2+10x-1

f'=30x^3+10x^2

Correct answer:

f'=30x^2+10x

Explanation:

To find the derivative of the function, we take the derivative of each element in the function independently, then add them up.

Using $\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$ , we solve

$\frac{\mathrm{d} }{\mathrm{d} x}(10x^3)+\frac{\mathrm{d} }{\mathrm{d} x}(5x^2)+\frac{\mathrm{d} }{\mathrm{d} x}(-1)=30x^2+10x$

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Example Question #60 : Derivative Rules For Sums, Products, And Quotients Of Functions

Determine the second derivative of f(x)=ln(x^2-1)

Possible Answers:

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

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

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

$f''(x)=\frac{2x^2-2}{x^3-x^2+1}$

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

Correct answer:

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

Explanation:

Finding our second derivative requires two steps, we first must find the derivative then find the corresponding rate of change for that new equation.

f(x)=ln(x^2-1)

Here, the chain rule is used since our function is of the form [f(g(x))]dx=f'(x)g'(f(x))

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

We now must use the quotient rule since our function is a rational function. We use the rule $[\frac{f(x)}{g(x)}]dx=\frac{f'(x)g(x)-g'(x)f(x))}{(g(x))^2}$

Therefore,

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

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Example Question #741 : Ap Calculus Ab

What is the derivative of x^3+2x+5 ?

Possible Answers:

$\frac{1}{3}x^2+\frac{1}{2}x+\frac{1}{5}$

3x^2+2

3x^2+2x

Correct answer:

3x^2+2

Explanation:

To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.

We're going to treat as 5x^0 , as anything to the zero power is one.

$\frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x+5)=(3*x^{3-1})+(1*2x^{1-1})+(0*5x^{0-1})$

Notice that $(0*5x^{0-1})=0$ , as anything times zero is zero.

$\frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x+5)=(3*x^{3-1})+(1*2x^{1-1})$

$\frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x+5)=(3*x^{2})+(2x^{0})$

$\frac{\mathrm{d} }{\mathrm{d} x}(x^3+2x+5)=3x^2+2$

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Example Question #61 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the slope of the $f(x)=\sqrt{x^2 +2x-3}$ at x=2 .

Possible Answers:

$6$

$\frac{6\sqrt{5}}{5}$

$\frac{\sqrt{5}}{5}$

$5$

$\frac{3\sqrt{5}}{5}$

Correct answer:

$\frac{3\sqrt{5}}{5}$

Explanation:

First we need to find the derivative of the function. $f'(x)=\frac{1+x}{\sqrt{-3+2x+x^2}}$

Now, we can plug in x=2 to the derivative function.

$f'(2)=\frac{3\sqrt{5}}{5}$

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Example Question #62 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following function:

$f=\frac{x^4\sin{x}}{x^2+3}$

Possible Answers:

$f'=\frac{(x^2+3)(4x^3\sin(x)+x^4\cos(x)-2x^5\sin(x))}{(x^2-3)^2}$

$f'=\frac{(x^2+3)(4x^3\sin(x)+x^4\cos(x)-2x^5\sin(x))}{(x^2+3)^2}$

$f'=\frac{(x^2+3)(4x^3\sin(x)+x^4\cos(x)-2x^5\cos(x))}{(x^2+3)^2}$

$f'=\frac{(x^2-3)(4x^3\sin(x)+x^4\cos(x)+2x^5\sin(x))}{(x^2+3)^2}$

Correct answer:

$f'=\frac{(x^2+3)(4x^3\sin(x)+x^4\cos(x)-2x^5\sin(x))}{(x^2+3)^2}$

Explanation:

To find the derivative of the function, we use a combination of the quotient rule and product rule

$f=\frac{x^4\sin{x}}{x^2+3}$

$f'=\frac{(x^2+3)(x^4\sin{x})'-x^4\sin{x}(x^2+3)'}{(x^2+3)^2}$

$f'=\frac{(x^2+3)(4x^3\sin{x}+x^4\cos{x})-x^4\sin{x}(2x)}{(x^2+3)^2}$

$f'=\frac{(x^2+3)(4x^3\sin(x)+x^4\cos(x)-2x^5\sin(x))}{(x^2+3)^2}$

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Example Question #63 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the function

f=x^3+5x^2+8x+20

Possible Answers:

f'=3x^2-10x+8

f'=3x^3+10x^2+8x

$f'=\frac{3x^2+10x^2+8}{x}$

f'=3x^2+10x+8

Correct answer:

f'=3x^2+10x+8

Explanation:

To find the derivative of the function, we use the rule

$\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$ and apply it to each term in the function

$f'=\frac{\mathrm{d} }{\mathrm{d} x}(x^3)+\frac{\mathrm{d} }{\mathrm{d} x}(5x^2)+\frac{\mathrm{d} }{\mathrm{d} x}(8x)+\frac{\mathrm{d} }{\mathrm{d} x}(20)$

f'=3x^2+10x+8

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Example Question #64 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following function

$f=\frac{x^3}{\tan{x}}$

Possible Answers:

$f'=\frac{3x^2\tan{x}-x^3\sec^2{x}}{\tan{x}}$

$f'=\frac{x^3\tan{x}-x^3\sec^2{x}}{\tan^2{x}}$

$f'=\frac{3x^2\tan{x}-x^3\sec^2{x}}{\tan^2{x}}$

$f'=\frac{3x^2\tan{x}+x^3\sec^2{x}}{\tan^2{x}}$

Correct answer:

$f'=\frac{3x^2\tan{x}-x^3\sec^2{x}}{\tan^2{x}}$

Explanation:

To find the derivative of the function, we use the quotient rule, which is

$\frac{\mathrm{d} }{\mathrm{d} x}(\frac{f}{g})=\frac{gf'-fg'}{g^2}$

Applying this to the function from the problem statement, we get

$f'=\frac{(x^3)'\tan{x}-x^3(\tan{x})'}{\tan^2{x}}$

$f'=\frac{3x^2\tan{x}-x^3\sec^2{x}}{\tan^2{x}}$

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Example Question #65 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the function

$f=x\sin{x}\cos{x}$

Hint: $\cos^2{x}-\sin^2{x}=\cos{2x}$

Possible Answers:

$f'=\sin{x}\cos{x}+x\cos{2x}$

$f'=\sin{x}\cos{x}+\cos{2x}$

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

$f'=\sin{x}\cos{x}+x\cos{x}$

Correct answer:

$f'=\sin{x}\cos{x}+x\cos{2x}$

Explanation:

To find the derivative of the function, we use the product rule, which by definition is

$\frac{\mathrm{d} }{\mathrm{d} x}(f*g)=f'g+g'f$

In this case, we can split up the product such that

$f=(x\sin{x})*(\cos{x})$

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

$y'=(\sin{x+x\cos{x}})(\cos{x})+\sin{x}(x\sin{x})$

$y'=\sin{x\cos{x}}+x\cos^2{x}+x\sin^2{x}$

$f'=\sin{x}\cos{x}+x\cos{2x}$

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Example Question #66 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the function:

y=x^3e^x

Possible Answers:

y'=3x^2e^x

y'=-e^x(x^3+3x^2)

y'=e^x(x^3+3x^2)

$y'=\frac{e^x}{(x^3+3x^2)}$

$y'=\frac{-e^x}{(x^3+3x^2)}$

Correct answer:

y'=e^x(x^3+3x^2)

Explanation:

This question requires us to understand two things.

First, the derivative of e^x is always e^x , such that the exponent contains a single variable (any other operation or numerical factors could cause the chain rule to come into play; a later topic)

Second, we must understand the product rule for derivatives. The product rule works as follows:

$\frac{dy}{dx}f(x)*g(x)=f(x)g'(x)+g(x)f'(x)$

Understanding these two concepts allows us to tackle the derivative of the given function.

y=x^3e^x

y'=(x^3)(e^x) +(e^x)(3x^2)

y'=e^x(x^3) +e^x(3x^2)

This simplifies to:

y'=e^x(x^3+3x^2)

We are finished taking the derivative of the product!

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AP Calculus AB : Computation of the Derivative

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #58 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #52 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #60 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #741 : Ap Calculus Ab

Example Question #61 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #62 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #63 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #64 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #65 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #66 : Derivative Rules For Sums, Products, And Quotients Of Functions

All AP Calculus AB Resources

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