Calculus Examples

$y = \frac{1}{x^{9}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

Rewrite $\frac{1}{x^{9}}$ as ${(x^{9})}^{- 1}$ .

$\frac{d}{d x} [{(x^{9})}^{- 1}]$

Step 1.2

Multiply the exponents in ${(x^{9})}^{- 1}$ .

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Step 1.2.1

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

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

Step 1.2.2

Multiply $9$ by $- 1$ .

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

Step 2

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

$- 9 x^{- 10}$

Step 3

Simplify.

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Step 3.1

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

$- 9 \frac{1}{x^{10}}$

Step 3.2

Combine terms.

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Step 3.2.1

Combine $- 9$ and $\frac{1}{x^{10}}$ .

$\frac{- 9}{x^{10}}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{9}{x^{10}}$