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AP Calculus AB : Derivatives

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Example Questions

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 #51 : 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'=4x+4

f'=4x+5

f'=9x

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 #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 #5 : Finding Derivative Of A Function

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

Possible Answers:

3x^2+2x

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

3x^2+2

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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AP Calculus AB : Derivatives

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

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 #51 : 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 #52 : Derivative Rules For Sums, Products, And Quotients Of Functions

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

Example Question #5 : Finding Derivative Of A Function

All AP Calculus AB Resources

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