Calculus Examples

$y = \frac{- 7}{\sqrt[3]{x}}$

Step 1

Simplify the expression.

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Step 1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{7}{\sqrt[3]{x}}]$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\frac{d}{d x} [- \frac{7}{x^{\frac{1}{3}}}]$

Step 2

Since $- 7$ is constant with respect to $x$ , the derivative of $- \frac{7}{x^{\frac{1}{3}}}$ with respect to $x$ is $- 7 \frac{d}{d x} [\frac{1}{x^{\frac{1}{3}}}]$ .

$- 7 \frac{d}{d x} [\frac{1}{x^{\frac{1}{3}}}]$

Step 3

Apply basic rules of exponents.

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Step 3.1

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

$- 7 \frac{d}{d x} [{(x^{\frac{1}{3}})}^{- 1}]$

Step 3.2

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

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Step 3.2.1

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

$- 7 \frac{d}{d x} [x^{\frac{1}{3} \cdot - 1}]$

Step 3.2.2

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

$- 7 \frac{d}{d x} [x^{\frac{- 1}{3}}]$

Step 3.2.3

Move the negative in front of the fraction.

$- 7 \frac{d}{d x} [x^{- \frac{1}{3}}]$

Step 4

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

$- 7 (- \frac{1}{3} x^{- \frac{1}{3} - 1})$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 7 (- \frac{1}{3} x^{- \frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 6

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

$- 7 (- \frac{1}{3} x^{- \frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Step 7

Combine the numerators over the common denominator.

$- 7 (- \frac{1}{3} x^{\frac{- 1 - 1 \cdot 3}{3}})$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $3$ .

$- 7 (- \frac{1}{3} x^{\frac{- 1 - 3}{3}})$

Step 8.2

Subtract $3$ from $- 1$ .

$- 7 (- \frac{1}{3} x^{\frac{- 4}{3}})$

Step 9

Move the negative in front of the fraction.

$- 7 (- \frac{1}{3} x^{- \frac{4}{3}})$

Step 10

Combine $x^{- \frac{4}{3}}$ and $\frac{1}{3}$ .

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

Step 11

Multiply $- 1$ by $- 7$ .

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

Step 12

Combine $7$ and $\frac{x^{- \frac{4}{3}}}{3}$ .

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

Step 13

Move $x^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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