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AP Calculus AB : Derivative rules for sums, products, and quotients of functions

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Example Questions

Example Question #51 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of y=5x^3e^x

Possible Answers:

15x^2e^x+5x^3

15x^2e^x

15x^2e^x+5x^3xe

5x^2e^x(3+x)

none of these answers

Correct answer:

5x^2e^x(3+x)

Explanation:

So whenever you have two distinct functions that are multiplied by each other, you will be using the product rule. So when looking at a function, see if you can separate it into two. In this case, we can see there is the function:

5x^3 and the function e^x .

So lets call those f(x) and g(x)

Then the product rule is that the derivative of f(x)g(x ) is:

f'(x)g(x)+f(x)g'(x) .

Then calculate to find:

f'(x)=15x^2 and g'(x)=e^x to give an answer of:

f'(x)g(x)+f(x)g'(x)=15x^2e^x+5x^3e^x

which simplifies to:

5x^2e^x(3+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=2x^3+5x^2+x-4

Possible Answers:

f'=6x^3+10x^2+x

f'=6x^2+11

f'=6x^2+10x+x-4

f'=6x^2+10x+1

Correct answer:

f'=6x^2+10x+1

Explanation:

To find the derivative of the sum, we take the derivative of each term independently, then add them all up. Further, we use the rule

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

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

f'=6x^2+10x+1

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Example Question #53 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following function

$f=5x\cos(x)$

Possible Answers:

$f'=5\cos(x)+5x\sin(x)$

$f'=5\cos(x)$

$f'=5\cos(x)+5\sin(x)$

$f'=5\cos(x)-5x\sin(x)$

Correct answer:

$f'=5\cos(x)-5x\sin(x)$

Explanation:

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

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

Using the function from the problem statement, we get

$f'=\frac{\mathrm{d} }{\mathrm{d} x}(5x)*(\cos(x))+\frac{\mathrm{d} }{\mathrm{d} x}(\cos(x))*5x$

$f'=5\cos(x)-5x\sin(x)$

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Example Question #54 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following function:

$f=\frac{4x^3+\sin(x)}{\tan(x)}$

Possible Answers:

$f'=\frac{\tan(x)(12x^3+\cos(x))-(4x^3+\cos(x))(\sec^2(x))}{\tan^2(x)}$

$f'=\frac{\tan(x)(12x^2+\cos(x))-(4x^3+\sin(x))(\sec^2(x))}{\sec^2(x)}$

$f'=\frac{\tan(x)(12x^2+\cos(x))-(4x^3+\sin(x))(\sec^2(x))}{\tan^2(x)}$

$f'=\frac{\tan(x)(12x^2+\cos(x))+(4x^3+\sin(x))(\tan^2(x))}{\tan^2(x)}$

Correct answer:

$f'=\frac{\tan(x)(12x^2+\cos(x))-(4x^3+\sin(x))(\sec^2(x))}{\tan^2(x)}$

Explanation:

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

$\frac{\mathrm{d} }{\mathrm{d} x}(\frac{f}{g})=\frac{gf'-fg'}{g^2}$ , where and are any expression

Using the function from the problem statement, we get

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

Taking the derivatives, we get

$f'=\frac{\tan(x)(12x^2+\cos(x))-(4x^3+\sin(x))(\sec^2(x))}{\tan^2(x)}$

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Example Question #55 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the function

$f=12x^2+\sin(x)-100$

Possible Answers:

$24x-\cos(x)$

$12x+\cos^2(x)$

$24x+\cos(x)$

$24x^2-\cos(x)$

Correct answer:

$24x+\cos(x)$

Explanation:

To find the derivative of the sum, we take the derivative of each term independently, then add them all up. Further, we use the rule

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

$f'=\frac{\mathrm{d} }{\mathrm{d} x}(12x^2)+\frac{\mathrm{d} }{\mathrm{d} x}(\sin(x))-\frac{\mathrm{d} }{\mathrm{d} x}(100)$

$f'=24x+\cos(x)$

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Example Question #56 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following function:

f=2x^2+5x-1

Possible Answers:

f'=4x^3+5x^2

f'=9x

f'=4x+5

f'=4x+4

Correct answer:

f'=4x+5

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

$\frac{\mathrm{d} }{\mathrm{d} x}(2x^2)+\frac{\mathrm{d} }{\mathrm{d} x}(5x)+\frac{\mathrm{d} }{\mathrm{d} x}(-1)=4x+5$

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Example Question #57 : Derivative Rules For Sums, Products, And Quotients Of Functions

Find the derivative of the following function:

$f=2x^5\sin{x}$

Possible Answers:

$f'=10x^5\sin{x}+x^5\cos{x}$

$f'=10x^4\sin{x}+2x^5\cos{x}$

$f'=10x^4\sin{x}+2x^5\sin{x}$

$f'=10x^4\sin{x}-2x^5\cos{x}$

Correct answer:

$f'=10x^4\sin{x}+2x^5\cos{x}$

Explanation:

To find the derivative of the function, we use the product rule, which is defined as

$\frac{\mathrm{d} }{\mathrm{d} x}(f*g)=(f'*g)+(f*g')$ , where f and g are both functions.

$\frac{\mathrm{d} }{\mathrm{d} x}(2x^5\sin{x})=\frac{\mathrm{d} }{\mathrm{d} x}(2x^5)*(\sin{x})+\frac{\mathrm{d} }{\mathrm{d} x}(\sin{x})*(2x^5)=10x^4\sin{x}+2x^5\cos{x}$

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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 #59 : 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^3+10x^2

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

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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AP Calculus AB : Derivative rules for sums, products, and quotients of functions

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #51 : Derivative Rules For Sums, Products, And Quotients Of Functions

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

Example Question #53 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #54 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #55 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #56 : Derivative Rules For Sums, Products, And Quotients Of Functions

Example Question #57 : Derivative Rules For Sums, Products, And Quotients Of Functions

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

Example Question #59 : Derivative Rules For Sums, Products, And Quotients Of Functions

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

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