Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

$3 (4 x^{3})$

Step 1.3

Multiply $4$ by $3$ .

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

$12 (3 x^{2})$

Step 2.3

Multiply $3$ by $12$ .

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

Step 3

Find the third derivative.

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Step 3.1

Since $36$ is constant with respect to $x$ , the derivative of $36 x^{2}$ with respect to $x$ is $36 \frac{d}{d x} [x^{2}]$ .

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

Step 3.2

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

$36 (2 x)$

Step 3.3

Multiply $2$ by $36$ .

$f''' (x) = 72 x$

Step 4

Find the fourth derivative.

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Step 4.1

Since $72$ is constant with respect to $x$ , the derivative of $72 x$ with respect to $x$ is $72 \frac{d}{d x} [x]$ .

$72 \frac{d}{d x} [x]$

Step 4.2

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

$72 \cdot 1$

Step 4.3

Multiply $72$ by $1$ .

$f^{4} (x) = 72$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $72$ .

$72$