# How do you find the derivative of sqrt(x^5)?

Jan 15, 2017

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

#### Explanation:

Rewrite the function using all fractional exponents and with knowledge of the rule ${\left({x}^{a}\right)}^{b} = {x}^{a b}$.

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

We now use the power rule, which says that the derivative of ${x}^{n}$ is equal to $\frac{d}{\mathrm{dx}} \left({x}^{n}\right) = n {x}^{n - 1}$.

So, the derivative of ${x}^{\frac{5}{2}}$ is:

$\frac{d}{\mathrm{dx}} \left({x}^{\frac{5}{2}}\right) = \frac{5}{2} {x}^{\frac{5}{2}} = \frac{5}{2} {x}^{\frac{3}{2}}$

Using the notation you used originally, you may write this as:

$\frac{d}{\mathrm{dx}} \left({x}^{\frac{5}{2}}\right) = \frac{5}{2} \sqrt{{x}^{3}}$