Calculus Examples

$\sqrt{49 x^{- 9}}$

Step 1

Rewrite $\frac{d}{d x} [\sqrt{49 x^{- 9}}]$ as $\frac{d}{d x} [7 \sqrt{x^{- 9}}]$ .

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Step 1.1

Rewrite $49$ as $7^{2}$ .

$\frac{d}{d x} [\sqrt{7^{2} x^{- 9}}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [7 \sqrt{x^{- 9}}]$

Step 2

Simplify the expression.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{- 9}}$ as $x^{\frac{- 9}{2}}$ .

$\frac{d}{d x} [7 x^{\frac{- 9}{2}}]$

Step 2.2

Move the negative in front of the fraction.

$\frac{d}{d x} [7 x^{- \frac{9}{2}}]$

Step 3

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{- \frac{9}{2}}$ with respect to $x$ is $7 \frac{d}{d x} [x^{- \frac{9}{2}}]$ .

$7 \frac{d}{d x} [x^{- \frac{9}{2}}]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{9}{2}$ .

$7 (- \frac{9}{2} x^{- \frac{9}{2} - 1})$

Step 5

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

$7 (- \frac{9}{2} x^{- \frac{9}{2} - 1 \cdot \frac{2}{2}})$

Step 6

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

$7 (- \frac{9}{2} x^{- \frac{9}{2} + \frac{- 1 \cdot 2}{2}})$

Step 7

Combine the numerators over the common denominator.

$7 (- \frac{9}{2} x^{\frac{- 9 - 1 \cdot 2}{2}})$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$7 (- \frac{9}{2} x^{\frac{- 9 - 2}{2}})$

Step 8.2

Subtract $2$ from $- 9$ .

$7 (- \frac{9}{2} x^{\frac{- 11}{2}})$

Step 9

Move the negative in front of the fraction.

$7 (- \frac{9}{2} x^{- \frac{11}{2}})$

Step 10

Combine $x^{- \frac{11}{2}}$ and $\frac{9}{2}$ .

$7 (- \frac{x^{- \frac{11}{2}} \cdot 9}{2})$

Step 11

Multiply $- 1$ by $7$ .

$- 7 \frac{x^{- \frac{11}{2}} \cdot 9}{2}$

Step 12

Combine $- 7$ and $\frac{x^{- \frac{11}{2}} \cdot 9}{2}$ .

$\frac{- 7 (x^{- \frac{11}{2}} \cdot 9)}{2}$

Step 13

Simplify the expression.

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Step 13.1

Multiply $9$ by $- 7$ .

$\frac{- 63 x^{- \frac{11}{2}}}{2}$

Step 13.2

Move $x^{- \frac{11}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 63}{2 x^{\frac{11}{2}}}$

Step 13.3

Move the negative in front of the fraction.

$- \frac{63}{2 x^{\frac{11}{2}}}$