Calculus Examples

$\frac{- 1}{3 x^{3}}$

Step 1

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

Apply basic rules of exponents.

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Step 3.1

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

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

Step 3.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 3.2.1

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

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

Step 3.2.2

Multiply $3$ by $- 1$ .

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

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$- \frac{1}{3} (- 3 x^{- 4})$

Step 5

Simplify terms.

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Step 5.1

Multiply $- 3$ by $- 1$ .

$3 (\frac{1}{3}) x^{- 4}$

Step 5.2

Combine $3$ and $\frac{1}{3}$ .

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

Step 5.3

Combine $\frac{3}{3}$ and $x^{- 4}$ .

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

Step 5.4

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{3 x^{4}}$

Step 5.5

Cancel the common factor of $3$ .

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Step 5.5.1

Cancel the common factor.

$\frac{3}{3 x^{4}}$

Step 5.5.2

Rewrite the expression.

$\frac{1}{x^{4}}$