Calculus Examples

$f (x) = 75 (1 - \frac{2}{x})$

Step 1

Since $75$ is constant with respect to $x$ , the derivative of $75 (1 - \frac{2}{x})$ with respect to $x$ is $75 \frac{d}{d x} [1 - \frac{2}{x}]$ .

$75 \frac{d}{d x} [1 - \frac{2}{x}]$

Step 2

By the Sum Rule, the derivative of $1 - \frac{2}{x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{2}{x}]$ .

$75 (\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{2}{x}])$

Step 3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$75 (0 + \frac{d}{d x} [- \frac{2}{x}])$

Step 4

Add $0$ and $\frac{d}{d x} [- \frac{2}{x}]$ .

$75 \frac{d}{d x} [- \frac{2}{x}]$

Step 5

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2}{x}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{1}{x}]$ .

$75 (- 2 \frac{d}{d x} [\frac{1}{x}])$

Step 6

Simplify the expression.

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Step 6.1

Multiply $- 2$ by $75$ .

$- 150 \frac{d}{d x} [\frac{1}{x}]$

Step 6.2

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

$- 150 \frac{d}{d x} [x^{- 1}]$

Step 7

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

$- 150 (- x^{- 2})$

Step 8

Multiply $- 1$ by $- 150$ .

$150 x^{- 2}$

Step 9

Simplify.

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Step 9.1

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

$150 \frac{1}{x^{2}}$

Step 9.2

Combine $150$ and $\frac{1}{x^{2}}$ .

$\frac{150}{x^{2}}$