How do you find the derivative of e^x * e^(2x) * e^(3x) * e^(4x)?

May 17, 2016

$\frac{d}{\mathrm{dx}} {e}^{x} \cdot {e}^{2 x} \cdot {e}^{3 x} \cdot {e}^{4 x} = 10 {e}^{10 x}$

Explanation:

Using the property that ${a}^{x} \cdot {a}^{y} = {a}^{x + y}$ along with that $\frac{d}{\mathrm{dx}} {e}^{x} = {e}^{x}$ and the chain rule, we have

$\frac{d}{\mathrm{dx}} {e}^{x} \cdot {e}^{2 x} \cdot {e}^{3 x} \cdot {e}^{4 x} = \frac{d}{\mathrm{dx}} {e}^{x + 2 x + 3 x + 4 x}$

$= \frac{d}{\mathrm{dx}} {e}^{10 x}$

$= {e}^{10 x} \left(\frac{d}{\mathrm{dx}} 10 x\right)$
(Using the chain rule with the functions ${e}^{x}$ and $10 x$)

$= 10 {e}^{10 x}$