Calculus Examples

$\frac{1}{\sqrt{x^{17}}}$

Step 1

Rewrite $\frac{d}{d x} [\frac{1}{\sqrt{x^{17}}}]$ as $\frac{d}{d x} [\frac{1}{x^{8} \sqrt{x}}]$ .

Tap for more steps...

Step 1.1

Rewrite $x^{17}$ as ${(x^{8})}^{2} x$ .

Tap for more steps...

Step 1.1.1

Factor out $x^{16}$ .

$\frac{d}{d x} [\frac{1}{\sqrt{x^{16} x}}]$

Step 1.1.2

Rewrite $x^{16}$ as ${(x^{8})}^{2}$ .

$\frac{d}{d x} [\frac{1}{\sqrt{{(x^{8})}^{2} x}}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [\frac{1}{x^{8} \sqrt{x}}]$

Step 2

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

$\frac{d}{d x} [\frac{1}{x^{8} x^{\frac{1}{2}}}]$

Step 3

Multiply $x^{8}$ by $x^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 3.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [\frac{1}{x^{8 + \frac{1}{2}}}]$

Step 3.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{d}{d x} [\frac{1}{x^{8 \cdot \frac{2}{2} + \frac{1}{2}}}]$

Step 3.3

Combine $8$ and $\frac{2}{2}$ .

$\frac{d}{d x} [\frac{1}{x^{\frac{8 \cdot 2}{2} + \frac{1}{2}}}]$

Step 3.4

Combine the numerators over the common denominator.

$\frac{d}{d x} [\frac{1}{x^{\frac{8 \cdot 2 + 1}{2}}}]$

Step 3.5

Simplify the numerator.

Tap for more steps...

Step 3.5.1

Multiply $8$ by $2$ .

$\frac{d}{d x} [\frac{1}{x^{\frac{16 + 1}{2}}}]$

Step 3.5.2

Add $16$ and $1$ .

$\frac{d}{d x} [\frac{1}{x^{\frac{17}{2}}}]$

Step 4

Apply basic rules of exponents.

Tap for more steps...

Step 4.1

Rewrite $\frac{1}{x^{\frac{17}{2}}}$ as ${(x^{\frac{17}{2}})}^{- 1}$ .

$\frac{d}{d x} [{(x^{\frac{17}{2}})}^{- 1}]$

Step 4.2

Multiply the exponents in ${(x^{\frac{17}{2}})}^{- 1}$ .

Tap for more steps...

Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d x} [x^{\frac{17}{2} \cdot - 1}]$

Step 4.2.2

Multiply $\frac{17}{2} \cdot - 1$ .

Tap for more steps...

Step 4.2.2.1

Combine $\frac{17}{2}$ and $- 1$ .

$\frac{d}{d x} [x^{\frac{17 \cdot - 1}{2}}]$

Step 4.2.2.2

Multiply $17$ by $- 1$ .

$\frac{d}{d x} [x^{\frac{- 17}{2}}]$

Step 4.2.3

Move the negative in front of the fraction.

$\frac{d}{d x} [x^{- \frac{17}{2}}]$

Step 5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{17}{2}$ .

$- \frac{17}{2} x^{- \frac{17}{2} - 1}$

Step 6

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

$- \frac{17}{2} x^{- \frac{17}{2} - 1 \cdot \frac{2}{2}}$

Step 7

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

$- \frac{17}{2} x^{- \frac{17}{2} + \frac{- 1 \cdot 2}{2}}$

Step 8

Combine the numerators over the common denominator.

$- \frac{17}{2} x^{\frac{- 17 - 1 \cdot 2}{2}}$

Step 9

Simplify the numerator.

Tap for more steps...

Step 9.1

Multiply $- 1$ by $2$ .

$- \frac{17}{2} x^{\frac{- 17 - 2}{2}}$

Step 9.2

Subtract $2$ from $- 17$ .

$- \frac{17}{2} x^{\frac{- 19}{2}}$

Step 10

Move the negative in front of the fraction.

$- \frac{17}{2} x^{- \frac{19}{2}}$

Step 11

Simplify.

Tap for more steps...

Step 11.1

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

$- \frac{17}{2} \cdot \frac{1}{x^{\frac{19}{2}}}$

Step 11.2

Combine terms.

Tap for more steps...

Step 11.2.1

Multiply $\frac{1}{x^{\frac{19}{2}}}$ by $\frac{17}{2}$ .

$- \frac{17}{x^{\frac{19}{2}} \cdot 2}$

Step 11.2.2

Move $2$ to the left of $x^{\frac{19}{2}}$ .

$- \frac{17}{2 x^{\frac{19}{2}}}$