Calculus Examples

$y = \frac{30}{x^{6}}$

Step 1

Since $30$ is constant with respect to $x$ , the derivative of $\frac{30}{x^{6}}$ with respect to $x$ is $30 \frac{d}{d x} [\frac{1}{x^{6}}]$ .

$30 \frac{d}{d x} [\frac{1}{x^{6}}]$

Step 2

Apply basic rules of exponents.

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Step 2.1

Rewrite $\frac{1}{x^{6}}$ as ${(x^{6})}^{- 1}$ .

$30 \frac{d}{d x} [{(x^{6})}^{- 1}]$

Step 2.2

Multiply the exponents in ${(x^{6})}^{- 1}$ .

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Step 2.2.1

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

$30 \frac{d}{d x} [x^{6 \cdot - 1}]$

Step 2.2.2

Multiply $6$ by $- 1$ .

$30 \frac{d}{d x} [x^{- 6}]$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 6$ .

$30 (- 6 x^{- 7})$

Step 4

Multiply $- 6$ by $30$ .

$- 180 x^{- 7}$

Step 5

Simplify.

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Step 5.1

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

$- 180 \frac{1}{x^{7}}$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $- 180$ and $\frac{1}{x^{7}}$ .

$\frac{- 180}{x^{7}}$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{180}{x^{7}}$