Calculus Examples

$f (x) = \frac{x^{5} - 5 x}{5}$

Step 1

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

$\frac{1}{5} \frac{d}{d x} [x^{5} - 5 x]$

Step 2

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

$\frac{1}{5} (\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5 x])$

Step 3

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

$\frac{1}{5} (5 x^{4} + \frac{d}{d x} [- 5 x])$

Step 4

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

$\frac{1}{5} (5 x^{4} - 5 \frac{d}{d x} [x])$

Step 5

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

$\frac{1}{5} (5 x^{4} - 5 \cdot 1)$

Step 6

Multiply $- 5$ by $1$ .

$\frac{1}{5} (5 x^{4} - 5)$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{1}{5} (5 x^{4}) + \frac{1}{5} \cdot - 5$

Step 7.2

Combine terms.

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Step 7.2.1

Combine $5$ and $\frac{1}{5}$ .

$\frac{5}{5} x^{4} + \frac{1}{5} \cdot - 5$

Step 7.2.2

Combine $\frac{5}{5}$ and $x^{4}$ .

$\frac{5 x^{4}}{5} + \frac{1}{5} \cdot - 5$

Step 7.2.3

Cancel the common factor of $5$ .

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Step 7.2.3.1

Cancel the common factor.

$\frac{5 x^{4}}{5} + \frac{1}{5} \cdot - 5$

Step 7.2.3.2

Divide $x^{4}$ by $1$ .

$x^{4} + \frac{1}{5} \cdot - 5$

Step 7.2.4

Combine $\frac{1}{5}$ and $- 5$ .

$x^{4} + \frac{- 5}{5}$

Step 7.2.5

Cancel the common factor of $- 5$ and $5$ .

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Step 7.2.5.1

Factor $5$ out of $- 5$ .

$x^{4} + \frac{5 \cdot - 1}{5}$

Step 7.2.5.2

Cancel the common factors.

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Step 7.2.5.2.1

Factor $5$ out of $5$ .

$x^{4} + \frac{5 \cdot - 1}{5 (1)}$

Step 7.2.5.2.2

Cancel the common factor.

$x^{4} + \frac{5 \cdot - 1}{5 \cdot 1}$

Step 7.2.5.2.3

Rewrite the expression.

$x^{4} + \frac{- 1}{1}$

Step 7.2.5.2.4

Divide $- 1$ by $1$ .

$x^{4} - 1$