Calculus Examples

$- \frac{60}{x^{9}}$

Step 1

Since $- 60$ is constant with respect to $x$ , the derivative of $- \frac{60}{x^{9}}$ with respect to $x$ is $- 60 \frac{d}{d x} [\frac{1}{x^{9}}]$ .

$- 60 \frac{d}{d x} [\frac{1}{x^{9}}]$

Step 2

Apply basic rules of exponents.

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Step 2.1

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

$- 60 \frac{d}{d x} [{(x^{9})}^{- 1}]$

Step 2.2

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

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Step 2.2.1

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

$- 60 \frac{d}{d x} [x^{9 \cdot - 1}]$

Step 2.2.2

Multiply $9$ by $- 1$ .

$- 60 \frac{d}{d x} [x^{- 9}]$

Step 3

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

$- 60 (- 9 x^{- 10})$

Step 4

Multiply $- 9$ by $- 60$ .

$540 x^{- 10}$

Step 5

Simplify.

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Step 5.1

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

$540 \frac{1}{x^{10}}$

Step 5.2

Combine $540$ and $\frac{1}{x^{10}}$ .

$\frac{540}{x^{10}}$