Calculus Examples

$f (x) = 12 x - x^{2} - \frac{3}{\sqrt{x^{2}}}$

Step 1

By the Sum Rule, the derivative of $12 x - x^{2} - \frac{3}{\sqrt{x^{2}}}$ with respect to $x$ is $\frac{d}{d x} [12 x] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- \frac{3}{\sqrt{x^{2}}}]$ .

$\frac{d}{d x} [12 x] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- \frac{3}{\sqrt{x^{2}}}]$

Step 2

Evaluate $\frac{d}{d x} [12 x]$ .

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Step 2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$12 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- \frac{3}{\sqrt{x^{2}}}]$

Step 2.2

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

$12 \cdot 1 + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- \frac{3}{\sqrt{x^{2}}}]$

Step 2.3

Multiply $12$ by $1$ .

$12 + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- \frac{3}{\sqrt{x^{2}}}]$

Step 3

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

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Step 3.1

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

$12 - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{3}{\sqrt{x^{2}}}]$

Step 3.2

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

$12 - (2 x) + \frac{d}{d x} [- \frac{3}{\sqrt{x^{2}}}]$

Step 3.3

Multiply $2$ by $- 1$ .

$12 - 2 x + \frac{d}{d x} [- \frac{3}{\sqrt{x^{2}}}]$

Step 4

Evaluate $\frac{d}{d x} [- \frac{3}{\sqrt{x^{2}}}]$ .

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Step 4.1

Pull terms out from under the radical, assuming positive real numbers.

$12 - 2 x + \frac{d}{d x} [- \frac{3}{x}]$

Step 4.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3}{x}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{1}{x}]$ .

$12 - 2 x - 3 \frac{d}{d x} [\frac{1}{x}]$

Step 4.3

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

$12 - 2 x - 3 \frac{d}{d x} [x^{- 1}]$

Step 4.4

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

$12 - 2 x - 3 (- x^{- 2})$

Step 4.5

Multiply $- 1$ by $- 3$ .

$12 - 2 x + 3 x^{- 2}$

Step 5

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

$12 - 2 x + 3 \frac{1}{x^{2}}$

Step 6

Simplify.

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Step 6.1

Combine $3$ and $\frac{1}{x^{2}}$ .

$12 - 2 x + \frac{3}{x^{2}}$

Step 6.2

Reorder terms.

$- 2 x + \frac{3}{x^{2}} + 12$