# How do you find the derivative of f(x)=5sqrtx?

May 11, 2016

$f ' \left(x\right) = \frac{5}{2 \sqrt{x}}$

#### Explanation:

Given,

$f \left(x\right) = 5 \sqrt{x}$

Rewrite the expression using $\frac{\mathrm{dy}}{\mathrm{dx}}$ notation.

$\frac{d}{\mathrm{dx}} \left(5 \sqrt{x}\right)$

Using the multiplication by a constant rule, $\left(c \cdot f\right) ' = c \cdot f '$, bring out the $5$.

$= 5 \cdot \frac{d}{\mathrm{dx}} \left(\sqrt{x}\right)$

Rewrite $\sqrt{x}$ using exponents.

$= 5 \cdot \frac{d}{\mathrm{dx}} \left({x}^{\frac{1}{2}}\right)$

Using the power rule, $\frac{d}{\mathrm{dx}} \left({x}^{n}\right) = n \cdot {x}^{n - 1}$, the expression becomes,

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

Simplify.

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

$= \frac{5}{2} {\left(\frac{1}{x}\right)}^{\frac{1}{2}}$

$= \frac{5}{2} \left(\frac{1}{\sqrt{x}}\right)$

$= \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\frac{5}{2 \sqrt{x}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$