Calculus Examples

$\frac{1}{4} x^{4} - 32 x^{2}$

Step 1

By the Sum Rule, the derivative of $\frac{1}{4} x^{4} - 32 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{4} x^{4}] + \frac{d}{d x} [- 32 x^{2}]$ .

$\frac{d}{d x} [\frac{1}{4} x^{4}] + \frac{d}{d x} [- 32 x^{2}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{1}{4} x^{4}]$ .

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Step 2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} x^{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{4} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 32 x^{2}]$

Step 2.2

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

$\frac{1}{4} (4 x^{3}) + \frac{d}{d x} [- 32 x^{2}]$

Step 2.3

Combine $4$ and $\frac{1}{4}$ .

$\frac{4}{4} x^{3} + \frac{d}{d x} [- 32 x^{2}]$

Step 2.4

Combine $\frac{4}{4}$ and $x^{3}$ .

$\frac{4 x^{3}}{4} + \frac{d}{d x} [- 32 x^{2}]$

Step 2.5

Cancel the common factor of $4$ .

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Step 2.5.1

Cancel the common factor.

$\frac{4 x^{3}}{4} + \frac{d}{d x} [- 32 x^{2}]$

Step 2.5.2

Divide $x^{3}$ by $1$ .

$x^{3} + \frac{d}{d x} [- 32 x^{2}]$

Step 3

Evaluate $\frac{d}{d x} [- 32 x^{2}]$ .

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Step 3.1

Since $- 32$ is constant with respect to $x$ , the derivative of $- 32 x^{2}$ with respect to $x$ is $- 32 \frac{d}{d x} [x^{2}]$ .

$x^{3} - 32 \frac{d}{d x} [x^{2}]$

Step 3.2

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

$x^{3} - 32 (2 x)$

Step 3.3

Multiply $2$ by $- 32$ .

$x^{3} - 64 x$