Calculus Examples

$f (x) = - 4 x^{2} (7 x^{2})$

Step 1

Find the first derivative.

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Step 1.1

Multiply $7$ by $- 4$ .

$\frac{d}{d x} [- 28 x^{2} x^{2}]$

Step 1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.2.1

Move $x^{2}$ .

$\frac{d}{d x} [- 28 (x^{2} x^{2})]$

Step 1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [- 28 x^{2 + 2}]$

Step 1.2.3

Add $2$ and $2$ .

$\frac{d}{d x} [- 28 x^{4}]$

Step 1.3

Since $- 28$ is constant with respect to $x$ , the derivative of $- 28 x^{4}$ with respect to $x$ is $- 28 \frac{d}{d x} [x^{4}]$ .

$- 28 \frac{d}{d x} [x^{4}]$

Step 1.4

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

$- 28 (4 x^{3})$

Step 1.5

Multiply $4$ by $- 28$ .

$f' (x) = - 112 x^{3}$

Step 2

Find the second derivative.

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Step 2.1

Since $- 112$ is constant with respect to $x$ , the derivative of $- 112 x^{3}$ with respect to $x$ is $- 112 \frac{d}{d x} [x^{3}]$ .

$- 112 \frac{d}{d x} [x^{3}]$

Step 2.2

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

$- 112 (3 x^{2})$

Step 2.3

Multiply $3$ by $- 112$ .

$f'' (x) = - 336 x^{2}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- 336 x^{2}$ .

$- 336 x^{2}$