Calculus Examples

$5 x^{2} - x^{3}$

Step 1

By the Sum Rule, the derivative of $5 x^{2} - x^{3}$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- x^{3}]$ .

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- x^{3}]$

Step 2

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x^{3}]$

Step 2.2

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

$5 (2 x) + \frac{d}{d x} [- x^{3}]$

Step 2.3

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [- x^{3}]$

Step 3

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

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$10 x - \frac{d}{d x} [x^{3}]$

Step 3.2

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

$10 x - (3 x^{2})$

Step 3.3

Multiply $3$ by $- 1$ .

$10 x - 3 x^{2}$

Step 4

Reorder terms.

$- 3 x^{2} + 10 x$