Calculus Examples

$- x^{\frac{8}{9}}$

Step 1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{8}{9}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{8}{9}}]$ .

$- \frac{d}{d x} [x^{\frac{8}{9}}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{8}{9}$ .

$- (\frac{8}{9} x^{\frac{8}{9} - 1})$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- (\frac{8}{9} x^{\frac{8}{9} - 1 \cdot \frac{9}{9}})$

Step 4

Combine $- 1$ and $\frac{9}{9}$ .

$- (\frac{8}{9} x^{\frac{8}{9} + \frac{- 1 \cdot 9}{9}})$

Step 5

Combine the numerators over the common denominator.

$- (\frac{8}{9} x^{\frac{8 - 1 \cdot 9}{9}})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $9$ .

$- (\frac{8}{9} x^{\frac{8 - 9}{9}})$

Step 6.2

Subtract $9$ from $8$ .

$- (\frac{8}{9} x^{\frac{- 1}{9}})$

Step 7

Move the negative in front of the fraction.

$- (\frac{8}{9} x^{- \frac{1}{9}})$

Step 8

Combine $\frac{8}{9}$ and $x^{- \frac{1}{9}}$ .

$- \frac{8 x^{- \frac{1}{9}}}{9}$

Step 9

Move $x^{- \frac{1}{9}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{8}{9 x^{\frac{1}{9}}}$