Calculus Examples

$f (x) = \frac{1}{12} x^{2} - 9 \sqrt{x}$

Step 1

By the Sum Rule, the derivative of $\frac{1}{12} x^{2} - 9 \sqrt{x}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{12} x^{2}] + \frac{d}{d x} [- 9 \sqrt{x}]$ .

$\frac{d}{d x} [\frac{1}{12} x^{2}] + \frac{d}{d x} [- 9 \sqrt{x}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{1}{12} x^{2}]$ .

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Step 2.1

Since $\frac{1}{12}$ is constant with respect to $x$ , the derivative of $\frac{1}{12} x^{2}$ with respect to $x$ is $\frac{1}{12} \frac{d}{d x} [x^{2}]$ .

$\frac{1}{12} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 \sqrt{x}]$

Step 2.2

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

$\frac{1}{12} (2 x) + \frac{d}{d x} [- 9 \sqrt{x}]$

Step 2.3

Combine $2$ and $\frac{1}{12}$ .

$\frac{2}{12} x + \frac{d}{d x} [- 9 \sqrt{x}]$

Step 2.4

Combine $\frac{2}{12}$ and $x$ .

$\frac{2 x}{12} + \frac{d}{d x} [- 9 \sqrt{x}]$

Step 2.5

Cancel the common factor of $2$ and $12$ .

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Step 2.5.1

Factor $2$ out of $2 x$ .

$\frac{2 (x)}{12} + \frac{d}{d x} [- 9 \sqrt{x}]$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $2$ out of $12$ .

$\frac{2 x}{2 \cdot 6} + \frac{d}{d x} [- 9 \sqrt{x}]$

Step 2.5.2.2

Cancel the common factor.

$\frac{2 x}{2 \cdot 6} + \frac{d}{d x} [- 9 \sqrt{x}]$

Step 2.5.2.3

Rewrite the expression.

$\frac{x}{6} + \frac{d}{d x} [- 9 \sqrt{x}]$

Step 3

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

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{x}{6} + \frac{d}{d x} [- 9 x^{\frac{1}{2}}]$

Step 3.2

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{\frac{1}{2}}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$\frac{x}{6} - 9 \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3.3

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

$\frac{x}{6} - 9 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{6} - 9 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 3.5

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{x}{6} - 9 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.6

Combine the numerators over the common denominator.

$\frac{x}{6} - 9 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

$\frac{x}{6} - 9 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 3.7.2

Subtract $2$ from $1$ .

$\frac{x}{6} - 9 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 3.8

Move the negative in front of the fraction.

$\frac{x}{6} - 9 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 3.9

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

$\frac{x}{6} - 9 \frac{x^{- \frac{1}{2}}}{2}$

Step 3.10

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

$\frac{x}{6} + \frac{- 9 x^{- \frac{1}{2}}}{2}$

Step 3.11

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{x}{6} + \frac{- 9}{2 x^{\frac{1}{2}}}$

Step 3.12

Move the negative in front of the fraction.

$\frac{x}{6} - \frac{9}{2 x^{\frac{1}{2}}}$