Calculus Examples

$y = x^{2} - x^{5}$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{5}$ with respect to $x$ is $- \frac{d}{d x} [x^{5}]$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$2 x - (5 x^{4})$

Step 2.3

Multiply $5$ by $- 1$ .

$2 x - 5 x^{4}$

Step 3

Reorder terms.

$- 5 x^{4} + 2 x$