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Precalculus : Find the Second Derivative of a Function

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Example Question #21 : Find The Second Derivative Of A Function

Find the second derivative of the function

$h(x)=e^{x}sin(x)$

Possible Answers:

$h''(x)=2x^{2}e^{x}-sin(x)$

$h''(x)=2e^{}xcos(x)$

$h''(x)=e^{x}+cos(x)$

$h''(x)=e^{x}sin(x)+e^{x}cos(x)$

$h''(x)=2e^{x}sin(x)$

Correct answer:

$h''(x)=2e^{}xcos(x)$

Explanation:

Use the product rule to get the first derivative.

Let $f(x)=e^{x}$ and g(x)=sin(x)

h'(x)=f'(x)g(x)+g'(x)f(x)

$h'(x)=(e^{x})'sin(x)+e^{x}(sin(x))'$

$h'(x)=e^{x}sin(x)+e^{x}cos(x)$

Use the product rule again for the second derivative.

$h''(x)=e^{x}sin(x)+e^{x}cos(x)+e^{x}cos(x)-e^{x}sin(x)$

$h''(x)=2e^{}xcos(x)$

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Example Question #41 : Derivatives

Find the second derivative of y=3x^4+5x^3 .

Possible Answers:

81x^3+15x^2

36x^4+30x^3

36x^3+30x^2

36x^2+30x

12x^3+15x^2

Correct answer:

36x^2+30x

Explanation:

To derive, use the power rule for derivatives.

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

Find the first derivative by taking the derivative of each term.

$y' = 4\cdot 3x^{4-1}+ 3\cdot 5x^{3-1} = 12x^3+15x^2$

Take the derivative of .

$y'' = 3\cdot 12x^{3-1}+2\cdot 15x^{2-1} = 36x^2+30x$

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Example Question #21 : Find The Second Derivative Of A Function

Determine the second derivative with the respect to x: $y=2e^{3x}$

Possible Answers:

$18 e^{9x}$

$18x^2e^{3x-3}-6xe^{3x-3}$

$18 e^{3x}$

18e^x

$18 e^{3x-3}$

Correct answer:

$18 e^{3x}$

Explanation:

To solve this, we first need to know the derivative of y=e^x with the respect to .

$\frac{d}{dx}e^x = e^x$

This problem will also involve the chain rule, which means that there we will need to take the derivative of the inner function inside e^x , since the power is not to the power of .

Find the first derivative.

$\frac{dy}{dx}= 2\cdot e^{3x} \cdot 3 = 6e^{3x}$

Find the second derivative by differentiating the first derivative.

$\frac{d^2y}{dx^2} = 6\cdot \e^{3x}\cdot 3 = 18 e^{3x}$

The answer is: $18 e^{3x}$

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Example Question #21 : Find The Second Derivative Of A Function

Find the second dervative for the following function.

f(x)=x^2+2x+5

Possible Answers:

2x+2

x^2+2x+5

Correct answer:

Explanation:

To find the second derivative, simply take the dervative twice according to the rules of derivatives.

f'(x)=2x+2

f''(x)=2

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Example Question #22 : Find The Second Derivative Of A Function

Find the second derivative of the following equation:

f(x)=0

Possible Answers:

undefined

Correct answer:

Explanation:

To solve, simply realize a constant derived is always 0. Thus, the answer is 0.

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Example Question #23 : Find The Second Derivative Of A Function

Find the second derivative of the following function:

f(x)=5x^3-8+3x^2

Possible Answers:

15x^2-6x

30x-6

15x^2+6x

30x+6

Correct answer:

30x+6

Explanation:

To solve, simply differentiate twice use the power rule, as outlined below.

Power rule: $(x^n)'=nx^{n-1}$

Also, remember the derivative of a constant is 0.

Thus,

$f'(x)=3*5x^{3-1}+6x^{2-1}=15x^2+6x$

$f''(x)=2*15x^{2-1}+6x^{1-1}=30x+6$

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Example Question #24 : Find The Second Derivative Of A Function

Find the second derivative of the function $f(x) = x\sin(x)$ .

Possible Answers:

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

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

$-x\sin(x)$

Correct answer:

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

Explanation:

To find the first derivative, apply the product rule, (f(x)*g(x))' = f'(x)*g(x) + g'(x)*f(x) , to the original function, where f(x) = x and g(x) = sin(x) , to get xcos(x) + $\sin(x)$ . We then apply the product rule to $x\cos(x)$ to arrive at $\cos (x) -x\sin (x)$ , and take the derivative of $\sin(x)$ to get $\cos(x)$ . Combining these two, we get the final answer, $2\cos(x) - x\sin (x)$ .

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Precalculus : Find the Second Derivative of a Function

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #21 : Find The Second Derivative Of A Function

Example Question #41 : Derivatives

Example Question #21 : Find The Second Derivative Of A Function

Example Question #21 : Find The Second Derivative Of A Function

Example Question #22 : Find The Second Derivative Of A Function

Example Question #23 : Find The Second Derivative Of A Function

Example Question #24 : Find The Second Derivative Of A Function

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