Calculus Examples

$f (x) = 5 - \frac{4}{x}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $5 - \frac{4}{x}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- \frac{4}{x}]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [- \frac{4}{x}]$

Step 1.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{4}{x}]$

Step 2

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

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Step 2.1

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

$0 - 4 \frac{d}{d x} [\frac{1}{x}]$

Step 2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$0 - 4 \frac{d}{d x} [x^{- 1}]$

Step 2.3

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

$0 - 4 (- x^{- 2})$

Step 2.4

Multiply $- 1$ by $- 4$ .

$0 + 4 x^{- 2}$

Step 3

Simplify.

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 + 4 \frac{1}{x^{2}}$

Step 3.2

Combine terms.

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Step 3.2.1

Combine $4$ and $\frac{1}{x^{2}}$ .

$0 + \frac{4}{x^{2}}$

Step 3.2.2

Add $0$ and $\frac{4}{x^{2}}$ .

$\frac{4}{x^{2}}$