Calculus Examples

$y = \sqrt[5]{x^{4}} + x^{4 e}$

Step 1

By the Sum Rule, the derivative of $\sqrt[5]{x^{4}} + x^{4 e}$ with respect to $x$ is $\frac{d}{d x} [\sqrt[5]{x^{4}}] + \frac{d}{d x} [x^{4 e}]$ .

$\frac{d}{d x} [\sqrt[5]{x^{4}}] + \frac{d}{d x} [x^{4 e}]$

Step 2

Evaluate $\frac{d}{d x} [\sqrt[5]{x^{4}}]$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{4}}$ as $x^{\frac{4}{5}}$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{5}$ .

$\frac{4}{5} x^{\frac{4}{5} - 1} + \frac{d}{d x} [x^{4 e}]$

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4}{5} x^{\frac{4}{5} - 1 \cdot \frac{5}{5}} + \frac{d}{d x} [x^{4 e}]$

Step 2.4

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{4}{5} x^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}} + \frac{d}{d x} [x^{4 e}]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{4}{5} x^{\frac{4 - 1 \cdot 5}{5}} + \frac{d}{d x} [x^{4 e}]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $5$ .

$\frac{4}{5} x^{\frac{4 - 5}{5}} + \frac{d}{d x} [x^{4 e}]$

Step 2.6.2

Subtract $5$ from $4$ .

$\frac{4}{5} x^{\frac{- 1}{5}} + \frac{d}{d x} [x^{4 e}]$

Step 2.7

Move the negative in front of the fraction.

$\frac{4}{5} x^{- \frac{1}{5}} + \frac{d}{d x} [x^{4 e}]$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4 e$ .

$\frac{4}{5} x^{- \frac{1}{5}} + 4 e x^{4 e - 1}$

Step 4

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

$\frac{4}{5} \cdot \frac{1}{x^{\frac{1}{5}}} + 4 e x^{4 e - 1}$

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

Multiply $\frac{4}{5}$ by $\frac{1}{x^{\frac{1}{5}}}$ .

$\frac{4}{5 x^{\frac{1}{5}}} + 4 e x^{4 e - 1}$

Step 5.1.2

To write $4 e x^{4 e - 1}$ as a fraction with a common denominator, multiply by $\frac{5 x^{\frac{1}{5}}}{5 x^{\frac{1}{5}}}$ .

$\frac{4}{5 x^{\frac{1}{5}}} + 4 e x^{4 e - 1} \cdot \frac{5 x^{\frac{1}{5}}}{5 x^{\frac{1}{5}}}$

Step 5.1.3

Combine $4 e x^{4 e - 1}$ and $\frac{5 x^{\frac{1}{5}}}{5 x^{\frac{1}{5}}}$ .

$\frac{4}{5 x^{\frac{1}{5}}} + \frac{4 e x^{4 e - 1} (5 x^{\frac{1}{5}})}{5 x^{\frac{1}{5}}}$

Step 5.1.4

Combine the numerators over the common denominator.

$\frac{4 + 4 e x^{4 e - 1} (5 x^{\frac{1}{5}})}{5 x^{\frac{1}{5}}}$

Step 5.1.5

Multiply $5$ by $4$ .

$\frac{4 + 20 e x^{4 e - 1} x^{\frac{1}{5}}}{5 x^{\frac{1}{5}}}$

Step 5.1.6

Multiply $x^{4 e - 1}$ by $x^{\frac{1}{5}}$ by adding the exponents.

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Step 5.1.6.1

Move $x^{\frac{1}{5}}$ .

$\frac{4 + 20 e (x^{\frac{1}{5}} x^{4 e - 1})}{5 x^{\frac{1}{5}}}$

Step 5.1.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 + 20 e x^{\frac{1}{5} + 4 e - 1}}{5 x^{\frac{1}{5}}}$

Step 5.1.6.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4 + 20 e x^{4 e + \frac{1}{5} - 1 \cdot \frac{5}{5}}}{5 x^{\frac{1}{5}}}$

Step 5.1.6.4

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{4 + 20 e x^{4 e + \frac{1}{5} + \frac{- 1 \cdot 5}{5}}}{5 x^{\frac{1}{5}}}$

Step 5.1.6.5

Combine the numerators over the common denominator.

$\frac{4 + 20 e x^{4 e + \frac{1 - 1 \cdot 5}{5}}}{5 x^{\frac{1}{5}}}$

Step 5.1.6.6

Simplify the numerator.

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Step 5.1.6.6.1

Multiply $- 1$ by $5$ .

$\frac{4 + 20 e x^{4 e + \frac{1 - 5}{5}}}{5 x^{\frac{1}{5}}}$

Step 5.1.6.6.2

Subtract $5$ from $1$ .

$\frac{4 + 20 e x^{4 e + \frac{- 4}{5}}}{5 x^{\frac{1}{5}}}$

Step 5.1.6.7

Move the negative in front of the fraction.

$\frac{4 + 20 e x^{4 e - \frac{4}{5}}}{5 x^{\frac{1}{5}}}$

Step 5.2

Reorder terms.

$\frac{20 e x^{4 e - \frac{4}{5}} + 4}{5 x^{\frac{1}{5}}}$