Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $x^{4} - 6$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 6]$

Step 1.2

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

$4 x^{3} + \frac{d}{d x} [- 6]$

Step 1.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$4 x^{3} + 0$

Step 1.4

Add $4 x^{3}$ and $0$ .

$f' (x) = 4 x^{3}$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

$4 (3 x^{2})$

Step 2.3

Multiply $3$ by $4$ .

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

Step 3

Find the third derivative.

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Step 3.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

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

Step 3.2

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

$12 (2 x)$

Step 3.3

Multiply $2$ by $12$ .

$f''' (x) = 24 x$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $24 x$ .

$24 x$