Calculus Examples

$y = x^{0.2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{0.2})$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 0.2$ .

$0.2 x^{- 0.8}$

Step 3.2

Simplify.

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Step 3.2.1

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

$0.2 \frac{1}{x^{0.8}}$

Step 3.2.2

Combine $0.2$ and $\frac{1}{x^{0.8}}$ .

$\frac{0.2}{x^{0.8}}$

Step 4

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

$y' = \frac{0.2}{x^{0.8}}$

Step 5

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

$\frac{d y}{d x} = \frac{0.2}{x^{0.8}}$