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High School Math : Finding Second Derivative of a Function

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Example Question #65 : Calculus I — Derivatives

What is the second derivative of $\sin(2x)$ ?

Possible Answers:

$-\sin(2x)$

$2\cos(2x)$

$-\sin(4x^2)$

$2\sin(2x)$

$-4\sin(2x)$

Correct answer:

$-4\sin(2x)$

Explanation:

To start we need to find the first derivative. For that, we need to use the chain rule: f(g(x))=f'(g(x)*g'(x)

$\frac{\mathrm{d} }{\mathrm{d} x}\sin(2x)=\frac{\mathrm{d} }{\mathrm{d} x} \sin(2x)*\frac{\mathrm{d} }{\mathrm{d} x}2x$

$\frac{\mathrm{d} }{\mathrm{d} x}\sin(2x)=\cos(2x)*2$

$\frac{\mathrm{d} }{\mathrm{d} x}\sin(2x)=2\cos(2x)$

Repeat the process!

$\frac{\mathrm{d} }{\mathrm{d} x}2\cos(2x)=\frac{\mathrm{d} }{\mathrm{d} x} 2\cos(2x)*\frac{\mathrm{d} }{\mathrm{d} x}2x$

Remember that $\frac{\mathrm{d} }{\mathrm{d} x}\cos(x)=-\sin(x)$ :

$\frac{\mathrm{d} }{\mathrm{d} x}2\cos(2x)=-2\sin(2x)*2$

$\frac{\mathrm{d} }{\mathrm{d} x}2\cos(2x)=-4\sin(2x)$

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Example Question #51 : Derivatives

What is the second derivative of x^4-3x^2+8 ?

Possible Answers:

12x^2-6

18x^2

4x^3-6x

12x

Correct answer:

12x^2-6

Explanation:

To find the second derivative, we need to start with the first derivative.

To find the first derivative of x^4-3x^2+8 , we can use the power rule.

The power rule states that we multiply each variable by its current exponent and then lower the exponent of each variable by one.

Since x^0=1 , we're going to treat as 8x^0 .

$\frac{\mathrm{d} }{\mathrm{d} x}x^4-3x^2+8x^0=(4x^{4-1})-(2*3x^{2-1})+(0*8x^{0-1})$

Anything times zero is zero, so our final term $(0*8x^{0-1})=0$ , regardless of the power of the exponent.

Simplify what we have.

$\frac{\mathrm{d} }{\mathrm{d} x}x^4-3x^2+8x^0=(4x^{3})-(6x^{1})$

$\frac{\mathrm{d} }{\mathrm{d} x}x^4-3x^2+8x^0=4x^{3}-6x$

Our first derivative, then, is 4x^3-6x .

To find the second derivative, we repeat the process using 4x^3-6x for our equation.

$\frac{\mathrm{d} }{\mathrm{d} x}4x^3-6x=(3*4x^{3-1})-(1*6x^{1-1})$

Simplify.

$\frac{\mathrm{d} }{\mathrm{d} x}4x^3-6x=(12x^{2})-(6x^{0})$

Remember that x^0=1 , which means our second derivative will be 12x^2-6 .

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Example Question #67 : Calculus I — Derivatives

If $f(x)=\frac{2}{3}x^3+7x^2-12x$ , what is f''(x) ?

Possible Answers:

$4x+4+\frac{12}{x}$

4x+14

$\frac{1}{6}x^4+\frac{7}{3}x^3-6x^2+c$

$2x^{2}+14x-12$

Correct answer:

4x+14

Explanation:

To find f''(x) , or the second derivative of our function, we need to start by finding the first derivative.

To find the first derivative, we can use the power rule. The power rule states that we multiply each variable by its current exponent and then lower that exponent by one.

$\frac{\mathrm{d} }{\mathrm{d} x}f(x)=(3*\frac{2}{3}x^{3-1})+(2*7x^{2-1})-(1*12x^{1-1})$

Simplify.

$\frac{\mathrm{d} }{\mathrm{d} x}f(x)=(2x^{2})+(14x^{1})-(12x^{0})$

Anything to the zero power is one, so 12x^0=12(1)=12 .

Therefore, $f'(x)=2x^{2}+14x-12$ .

Now we repeat the process, but we use $f'(x)=2x^{2}+14x-12$ .

$\frac{\mathrm{d} }{\mathrm{d} x}f'(x)=(2*2x^{2-1})+(1*14x^{1-1})-(0*12x^{0-1})$

Remember, anything times zero is zero.

$\frac{\mathrm{d} }{\mathrm{d} x}f'(x)=(4x^{1})+(14x^{0})-(0)$

f''(x)=4x+14

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High School Math : Finding Second Derivative of a Function

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #65 : Calculus I — Derivatives

Example Question #51 : Derivatives

Example Question #67 : Calculus I — Derivatives

All High School Math Resources

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