# How do you find the derivative of  [e^(1/2x)]/(2x^3)?

Nov 25, 2016

Use the rule for the derivative of the products of tho functions:

$d \frac{f \cdot g}{\mathrm{dx}} = g \cdot \frac{\mathrm{df}}{\mathrm{dx}} + f \cdot \frac{\mathrm{dg}}{\mathrm{dx}}$

#### Explanation:

$\frac{{e}^{\frac{1}{2} x}}{2 {x}^{3}} = \frac{1}{2} {x}^{- 3} {e}^{\frac{1}{2} x}$

$\frac{d}{\mathrm{dx}} \left(\frac{{e}^{\frac{1}{2} x}}{2 {x}^{3}}\right) = {e}^{\frac{1}{2} x} \cdot d \frac{\left(\frac{1}{2} {x}^{- 3}\right)}{\mathrm{dx}} + \frac{1}{2} {x}^{- 3} \cdot d \frac{{e}^{\frac{1}{2} x}}{\mathrm{dx}} =$

$= - \frac{3}{2} {x}^{- 4} {e}^{\frac{1}{2} x} + \frac{1}{4} {x}^{- 3} {e}^{\frac{1}{2} x} = {e}^{\frac{1}{2} x} / \left(4 {x}^{4}\right) \left(x - 6\right)$