Calculus Examples

$\frac{\sqrt[3]{64 x^{6}}}{x^{5}}$

Step 1

Rewrite $\frac{d}{d x} [\frac{\sqrt[3]{64 x^{6}}}{x^{5}}]$ as $\frac{d}{d x} [\frac{4 x^{2}}{x^{5}}]$ .

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Step 1.1

Rewrite $64 x^{6}$ as ${(4 x^{2})}^{3}$ .

$\frac{d}{d x} [\frac{\sqrt[3]{{(4 x^{2})}^{3}}}{x^{5}}]$

Step 1.2

Pull terms out from under the radical, assuming real numbers.

$\frac{d}{d x} [\frac{4 x^{2}}{x^{5}}]$

Step 2

Cancel the common factor of $x^{2}$ and $x^{5}$ .

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Step 2.1

Factor $x^{2}$ out of $4 x^{2}$ .

$\frac{d}{d x} [\frac{x^{2} \cdot 4}{x^{5}}]$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $x^{2}$ out of $x^{5}$ .

$\frac{d}{d x} [\frac{x^{2} \cdot 4}{x^{2} x^{3}}]$

Step 2.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{x^{2} \cdot 4}{x^{2} x^{3}}]$

Step 2.2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{4}{x^{3}}]$

Step 3

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{x^{3}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$4 \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

$4 \frac{d}{d x} [x^{3 \cdot - 1}]$

Step 4.2.2

Multiply $3$ by $- 1$ .

$4 \frac{d}{d x} [x^{- 3}]$

Step 5

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

$4 (- 3 x^{- 4})$

Step 6

Multiply $- 3$ by $4$ .

$- 12 x^{- 4}$

Step 7

Simplify.

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Step 7.1

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

$- 12 \frac{1}{x^{4}}$

Step 7.2

Combine terms.

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Step 7.2.1

Combine $- 12$ and $\frac{1}{x^{4}}$ .

$\frac{- 12}{x^{4}}$

Step 7.2.2

Move the negative in front of the fraction.

$- \frac{12}{x^{4}}$