Calculus Examples

$x^{4} - 4 x^{3}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{4} - 4 x^{3}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}]$ .

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

Step 1.2

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

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

Step 2

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

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

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

Step 2.2

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

$4 x^{3} - 4 (3 x^{2})$

Step 2.3

Multiply $3$ by $- 4$ .

$4 x^{3} - 12 x^{2}$