Calculus Examples

$y = x^{- \sqrt{5}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$y = x^{- 5^{\frac{1}{2}}}$

Step 2

Differentiate both sides of the equation.

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

Step 3

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

$y'$

Step 4

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

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

Step 5

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

$y' = - 5^{\frac{1}{2}} x^{- 5^{\frac{1}{2}} - 1}$

Step 6

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

$\frac{d y}{d x} = - 5^{\frac{1}{2}} x^{- 5^{\frac{1}{2}} - 1}$