Calculus Examples

$y = 2 \sqrt[5]{x^{2}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2}}$ as $x^{\frac{2}{5}}$ .

$y = 2 x^{\frac{2}{5}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (2 x^{\frac{2}{5}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{\frac{2}{5}}$ with respect to $x$ is $2 \frac{d}{d x} [x^{\frac{2}{5}}]$ .

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

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{5}$ .

$2 (\frac{2}{5} x^{\frac{2}{5} - 1})$

Step 4.3

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

$2 (\frac{2}{5} x^{\frac{2}{5} - 1 \cdot \frac{5}{5}})$

Step 4.4

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

$2 (\frac{2}{5} x^{\frac{2}{5} + \frac{- 1 \cdot 5}{5}})$

Step 4.5

Combine the numerators over the common denominator.

$2 (\frac{2}{5} x^{\frac{2 - 1 \cdot 5}{5}})$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $5$ .

$2 (\frac{2}{5} x^{\frac{2 - 5}{5}})$

Step 4.6.2

Subtract $5$ from $2$ .

$2 (\frac{2}{5} x^{\frac{- 3}{5}})$

Step 4.7

Move the negative in front of the fraction.

$2 (\frac{2}{5} x^{- \frac{3}{5}})$

Step 4.8

Combine $\frac{2}{5}$ and $x^{- \frac{3}{5}}$ .

$2 \frac{2 x^{- \frac{3}{5}}}{5}$

Step 4.9

Combine $2$ and $\frac{2 x^{- \frac{3}{5}}}{5}$ .

$\frac{2 (2 x^{- \frac{3}{5}})}{5}$

Step 4.10

Multiply.

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Step 4.10.1

Multiply $2$ by $2$ .

$\frac{4 x^{- \frac{3}{5}}}{5}$

Step 4.10.2

Move $x^{- \frac{3}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{5 x^{\frac{3}{5}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{4}{5 x^{\frac{3}{5}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{4}{5 x^{\frac{3}{5}}}$