Calculus Examples

$\frac{d^{2}}{d x^{2}} \cdot (3 x^{4} + 4 x^{2})$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $3 x^{4} + 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [4 x^{2}]$ .

$\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [4 x^{2}]$

Step 1.2

Evaluate $\frac{d}{d x} [3 x^{4}]$ .

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Step 1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [4 x^{2}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$3 (4 x^{3}) + \frac{d}{d x} [4 x^{2}]$

Step 1.2.3

Multiply $4$ by $3$ .

$12 x^{3} + \frac{d}{d x} [4 x^{2}]$

Step 1.3

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$12 x^{3} + 4 \frac{d}{d x} [x^{2}]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$12 x^{3} + 4 (2 x)$

Step 1.3.3

Multiply $2$ by $4$ .

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

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $12 x^{3} + 8 x$ with respect to $x$ is $\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [8 x]$ .

$\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [8 x]$

Step 2.2

Evaluate $\frac{d}{d x} [12 x^{3}]$ .

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Step 2.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{3}$ with respect to $x$ is $12 \frac{d}{d x} [x^{3}]$ .

$12 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [8 x]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$12 (3 x^{2}) + \frac{d}{d x} [8 x]$

Step 2.2.3

Multiply $3$ by $12$ .

$36 x^{2} + \frac{d}{d x} [8 x]$

Step 2.3

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 2.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$36 x^{2} + 8 \frac{d}{d x} [x]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$36 x^{2} + 8 \cdot 1$

Step 2.3.3

Multiply $8$ by $1$ .

$f'' (x) = 36 x^{2} + 8$