Calculus Examples

$f (x) = \frac{2}{3 x^{6}}$

Step 1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2}{3 x^{6}}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [\frac{1}{x^{6}}]$ .

$\frac{2}{3} \frac{d}{d x} [\frac{1}{x^{6}}]$

Step 2

Apply basic rules of exponents.

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Step 2.1

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

$\frac{2}{3} \frac{d}{d x} [{(x^{6})}^{- 1}]$

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, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2}{3} \frac{d}{d x} [x^{6 \cdot - 1}]$

Step 2.2.2

Multiply $6$ by $- 1$ .

$\frac{2}{3} \frac{d}{d x} [x^{- 6}]$

Step 3

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

$\frac{2}{3} (- 6 x^{- 7})$

Step 4

Simplify terms.

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Step 4.1

Combine $- 6$ and $\frac{2}{3}$ .

$\frac{- 6 \cdot 2}{3} x^{- 7}$

Step 4.2

Multiply $- 6$ by $2$ .

$\frac{- 12}{3} x^{- 7}$

Step 4.3

Combine $\frac{- 12}{3}$ and $x^{- 7}$ .

$\frac{- 12 x^{- 7}}{3}$

Step 4.4

Move $x^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 12}{3 x^{7}}$

Step 4.5

Cancel the common factor of $- 12$ and $3$ .

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Step 4.5.1

Factor $3$ out of $- 12$ .

$\frac{3 \cdot - 4}{3 x^{7}}$

Step 4.5.2

Cancel the common factors.

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Step 4.5.2.1

Factor $3$ out of $3 x^{7}$ .

$\frac{3 \cdot - 4}{3 (x^{7})}$

Step 4.5.2.2

Cancel the common factor.

$\frac{3 \cdot - 4}{3 x^{7}}$

Step 4.5.2.3

Rewrite the expression.

$\frac{- 4}{x^{7}}$

Step 4.6

Move the negative in front of the fraction.

$- \frac{4}{x^{7}}$