Calculus Examples

x^{4}

$x^{4}$

Step 1

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 4

$n = 4$ .

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

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

Find the fourth derivative.

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Step 4.1

Since

$24$ is constant with respect to

$x$ , the derivative of

$24 x$ with respect to

$x$ is

$24 \frac{d}{d x} [x]$ .

$24 \frac{d}{d x} [x]$

Step 4.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$24 \cdot 1$

Step 4.3

Multiply

$24$ by

$1$ .

$f^{4} (x) = 24$

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