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Precalculus : Find the Second Derivative of a Function

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Example Questions

Example Question #11 : Find The Second Derivative Of A Function

Find the second derivative of the following function:

f(x)=(4x^3+x)^2

Possible Answers:

f''(x)=120x^4+72x^2+1

f''(x)=960x^4+48x^2+6

f''(x)=480x^4+96x^2+2

f''(x)=240x^4+48x^2+4

f''(x)=60x^4+12x^2+3

Correct answer:

f''(x)=480x^4+96x^2+2

Explanation:

We could multiply out our function and then find the second derivative, or we could apply the chain rule to find the first derivative and then apply the product rule to find the second derivative. Let's try the second method:

f(x)=(4x^3+x)^2

f'(x)=2(4x^3+x)(12x^2+1)

Now that we have our first derivative from the chain rule, we can find the second derivative using the product rule:

f'(x)=2(4x^3+x)(12x^2+1)

f''(x)=2(12x^2+1)(12x^2+1)+2(4x^3+x)(24x)

Which we can then multiply out and simplify the arrive at the following answer:

f''(x)=480x^4+96x^2+2

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Example Question #12 : Find The Second Derivative Of A Function

Find the second derivative of $f(x)=x^{2}-4x+9$ .

Possible Answers:

f''(x)=2x-4

f''(x)=2x-5

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

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

f''(x)=2

Correct answer:

f''(x)=2

Explanation:

For any function $f(x)=Ax^{n}$ , the first derivative $f'(x)=nAx^{n-1}$ and the second derivative is $f''(x)=(n-1)(n)(An^{n-2})$

Therefore, taking each term of f(x) :

$f(x)=x^{2}-4x+9$

$f'(x)=(2)x^{2-1}-(1)(4x^{1-1})+(0)(9)$

f'(x)=2x-4

Then, taking the derivative again:

f'(x)=2x-4

$f''(x)=(1)(2x^{1-1})-(0)(4)$

f''(x)=2

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Example Question #13 : Find The Second Derivative Of A Function

Find the second derivative of the function $f(x)=2x^{4}-6x^{2}+9x+4$

Possible Answers:

$f''(x)=24x^{2}-12$

$f''(x)=8x^{3}-12x+9$

$f''(x)=8x^{3}-12$

$f''(x)=24x^{2}-12x$

f''(x)=24x-12

Correct answer:

$f''(x)=24x^{2}-12$

Explanation:

For any function $f(x)=Ax^{n}$ , the first derivative $f'(x)=nAx^{n-1}$ and the second derivative is $f''(x)=(n-1)(n)(An^{n-2})$ .

Taking the first derivative of f(x) :

$f(x)=2x^{4}-6x^{2}+9x+4$

$f'(x)=4(2x^{4-1})-(2)(6x^{2-1})+(1)(9x^{1-1})+(0)4$

$f'(x)=8x^{3}-12x+9$

Then, taking the second derivative:

$f'(x)=8x^{3}-12x+9$

$f''(x)=3(8x^{3-1})-(1)(12x^{1-1})+(0)9$

$f''(x)=24x^{2}-12$

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Example Question #14 : Find The Second Derivative Of A Function

Find the second derivative of the function $f(x)=x^{3}+2x^{2}-5x+4$

Possible Answers:

$f''(x)=3x^{2}+8x$

$f''(x)=3x^{2}+8x-5$

$f''(x)=6x^{2}+8x$

$f''(x)=6x^{2}+8$

f''(x)=6x+4

Correct answer:

f''(x)=6x+4

Explanation:

For any function $f(x)=Ax^{n}$ , the first derivative $f'(x)=nAx^{n-1}$ and the second derivative is $f''(x)=(n-1)(n)(An^{n-2})$ .

Taking the first derivative of f(x) :

$f(x)=x^{3}+2x^{2}-5x+4$

$f'(x)=3x^{3-1}+(2)(2x^{2-1})-(1)(5x^{1-1})+(0)4$

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

Then, taking the second derivative of f(x) :

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

$f''(x)=2(3x^{2-1})+(1)(4x^{1-1})-(0)5$

f''(x)=6x+4

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Example Question #11 : Find The Second Derivative Of A Function

Find the second dervative of the following equation.

f(x)=x^3+4x^2-7x+2

Possible Answers:

f''(x)=6+8x

f''(x)=x^3+4x^2

f''(x)=6x+8

f''(x)=3+8x

Correct answer:

f''(x)=6x+8

Explanation:

To find the second derivative, simply differentiate the equation twice.

To differentiate this problem we will need to use the power rule which states,

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$ .

Applying the power rule to each term twice, we are able to find the second derivative.

f(x)=x^3+4x^2-7x+2

f'(x)=3x^2+8x-7

f''(x)=6x+8

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Example Question #12 : Find The Second Derivative Of A Function

Find the second derivative of with respect to when

y=2x^6+3x^3-4x^2-5x+9 .

Possible Answers:

y''=60x^4+18x+8

y''=60x^4+18x-8

y''=13x^4+8x-6

y''=12x^5+9x^2-8x-5

Correct answer:

y''=60x^4+18x-8

Explanation:

For this problem we will need to use the power rule on each term.

The power rule is,

$y=x^n\rightarrow y'=nx^{n-1}$

Applying the power rule to our function we get the following derivative.

y=2x^6+3x^3-4x^2-5x+9

y'=12x^5+9x^2-8x-5+0

y''=60x^4+18x-8-0+0

y''=60x^4+18x-8

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Example Question #11 : Find The Second Derivative Of A Function

Find the second derivative of $y=\cos(x)$ .

Possible Answers:

$y''=-\sin(x)$

$y''=\cos(x)$

$y''=-\cos(x)$

$y''=\sin(x)$

Correct answer:

$y''=-\cos(x)$

Explanation:

We first need to find the first derivative of $y=\cos(x)$ . Remember that according to the derivatives of trigonometric functions, the derviative of cosine is negative sine and the derivative of sine is cosine.

Applying these rules we are able to find the first derivative.

$y'=-\sin(x)$

Now to find the second derivative we take the derivative of the first derivative.

$y''=-(\sin(x))'$

$y''=-(\cos(x))$

$y''=-\cos(x)$

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Example Question #703 : Pre Calculus

Find the second derivative of $f(x)=\ln(2+x^3)$ .

Possible Answers:

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

$f''(x)=\frac{6x}{2+x^3}$

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

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

Correct answer:

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

Explanation:

To find the second derivative of $f(x)=\ln(2+x^3)$ we will need to take the derivative of the first derivative.

To find the first derivative we will use the rule for natural logs which states,

$ln(u)'=\frac{1}{u}\cdot \frac{du}{dx}$

Applying this rule to our function we get the following.

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

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

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

Now that we have the first derivative we will take the derivative of it to get the second derivative. In order to do so we will need to use the quotient rule which states,

$\frac{f(x)}{g(x)}\rightarrow \frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}$

Applying this rule we get the following.

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

$f''(x)=\frac{6x(2+x^3)-(3x^2)(3x^2)}{(2+x^3)^2}$

$f''(x)=\frac{12x+6x^4-9x^4}{(2+x^3)^2}$

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

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

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Example Question #12 : Find The Second Derivative Of A Function

Find the second derivative of $y=\sin(x)\cos(x)$ .

Possible Answers:

$y''=2\cos(x)-2\sin(x)$

$y''=2\cos(x)-\sin(x)-2\sin(x)\cos(x)$

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

$y''=-4\sin(x)\cos(x)$

Correct answer:

$y''=-4\sin(x)\cos(x)$

Explanation:

To find the second derivative of this function we will need to use the following rules.

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

Product Rule,

$f(x)g(x)\rightarrow f(x)g'(x)+g(x)f'(x)$ .

Chain Rule,

$[f(g(x))]'\rightarrow f'(g(x))\cdot g'(x)$ .

Applying these rules we can find both the first derivative and then the second derivative.

$y=\sin(x)\cos(x)$

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

$y'=\cos(x)\cos(x)+\sin(x)(-\sin(x))$

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

$y''=2\cos(x)(\cos(x))'-2\sin(x)(\sin(x))'$

$y''=2\cos(x)(-\sin(x))-2\sin(x)\cos(x)$

$y''=-2\sin(x)\cos(x)-2\sin(x)\cos(x)$

$y''=-4\sin(x)\cos(x)$

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Example Question #13 : Find The Second Derivative Of A Function

Find the second derivative of: 3x^4+2x^3+2x^2+4

Possible Answers:

f''(x)=12x^3+6x^2+4

f''(x)=36x^2+12x+4

f''(x)=12x^3+6x^2+4x

f''(x)=36x^2+12x+4x

None of the other answers.

Correct answer:

f''(x)=36x^2+12x+4

Explanation:

The derivative of this function, known as f'(x) (read as f prime of x), can be found by using the Power rule of derivatives on each term in the function. Power rule: $x^n=nx^{n-1}$ . If there is a value in front of x we multiply it by the "n" we carried over. For example taking the derivative of 2x^3=6x^2 . To find the second derivative, simply repeat the process. f''(x) is read as "f double prime of x" or "the second derivative of f(x)".

f(x)=3x^4+2x^3+2x^2+4

f'(x)=12x^3+6x^2+4x

f''(x)=36x^2+12x+4

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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 #11 : Find The Second Derivative Of A Function

Example Question #12 : Find The Second Derivative Of A Function

Example Question #13 : Find The Second Derivative Of A Function

Example Question #14 : Find The Second Derivative Of A Function

Example Question #11 : Find The Second Derivative Of A Function

Example Question #12 : Find The Second Derivative Of A Function

Example Question #11 : Find The Second Derivative Of A Function

Example Question #703 : Pre Calculus

Example Question #12 : Find The Second Derivative Of A Function

Example Question #13 : Find The Second Derivative Of A Function

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