Calculus Examples

$12 - x - \frac{9}{x}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $12 - x - \frac{9}{x}$ with respect to $x$ is $\frac{d}{d x} [12] + \frac{d}{d x} [- x] + \frac{d}{d x} [- \frac{9}{x}]$ .

$\frac{d}{d x} [12] + \frac{d}{d x} [- x] + \frac{d}{d x} [- \frac{9}{x}]$

Step 1.2

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x] + \frac{d}{d x} [- \frac{9}{x}]$

Step 2

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

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Step 2.1

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

$0 - \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{9}{x}]$

Step 2.2

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

$0 - 1 \cdot 1 + \frac{d}{d x} [- \frac{9}{x}]$

Step 2.3

Multiply $- 1$ by $1$ .

$0 - 1 + \frac{d}{d x} [- \frac{9}{x}]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{9}{x}]$ .

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Step 3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- \frac{9}{x}$ with respect to $x$ is $- 9 \frac{d}{d x} [\frac{1}{x}]$ .

$0 - 1 - 9 \frac{d}{d x} [\frac{1}{x}]$

Step 3.2

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

$0 - 1 - 9 \frac{d}{d x} [x^{- 1}]$

Step 3.3

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

$0 - 1 - 9 (- x^{- 2})$

Step 3.4

Multiply $- 1$ by $- 9$ .

$0 - 1 + 9 x^{- 2}$

Step 4

Simplify.

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Step 4.1

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

$0 - 1 + 9 \frac{1}{x^{2}}$

Step 4.2

Combine terms.

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Step 4.2.1

Subtract $1$ from $0$ .

$- 1 + 9 \frac{1}{x^{2}}$

Step 4.2.2

Combine $9$ and $\frac{1}{x^{2}}$ .

$- 1 + \frac{9}{x^{2}}$

Step 4.3

Reorder terms.

$\frac{9}{x^{2}} - 1$