# What is the first derivative of a complex surge function?

## this is the complex surge function $y = k {x}^{a} {e}^{- b x}$ where a, b, k are positive constants.

Apr 14, 2018

$y ' \left(x\right) = a k {x}^{a - 1} {e}^{- b x} - b k {x}^{a} \cdot {e}^{- b x}$

#### Explanation:

use the product rule
[$\frac{d}{\mathrm{dx}} \left(f \left(x\right) g \left(x\right)\right) = f ' \left(x\right) g \left(x\right) + f \left(x\right) g ' \left(x\right)$]

$y ' \left(x\right) = \frac{d}{\mathrm{dx}} \left(k {x}^{a}\right) \cdot \left({e}^{- b x}\right) + k {x}^{a} \cdot \frac{d}{\mathrm{dx}} \left({e}^{- b x}\right)$
$y ' \left(x\right) = a k {x}^{a - 1} {e}^{- b x} + k {x}^{a} \cdot {e}^{- b x} \left(- b\right)$
$y ' \left(x\right) = a k {x}^{a - 1} {e}^{- b x} - b k {x}^{a} \cdot {e}^{- b x}$

could also equal:
$y ' \left(x\right) = k {x}^{a - 1} {e}^{- b x} \left(a - b x\right)$