# What is the derivative of (5x)/e^x?

Sep 5, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = 5 {e}^{-} x \left(1 - x\right)$.

#### Explanation:

Let $y = 5 \frac{x}{e} ^ x = 5 x \cdot {e}^{-} x$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{d}{\mathrm{dx}} \left(5 x \cdot {e}^{-} x\right) = 5 \frac{d}{\mathrm{dx}} \left(x \cdot {e}^{-} x\right)$.

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = 5 \left[x \cdot \frac{d}{\mathrm{dx}} \left({e}^{-} x\right) + {e}^{-} x \cdot \frac{d}{\mathrm{dx}} \left(x\right)\right]$

$= 5 \left\{x \cdot {e}^{-} x \cdot \frac{d}{\mathrm{dx}} \left(- x\right) + {e}^{-} x \cdot 1\right] \ldots \ldots \ldots \ldots . . \text{[Chain Rule]}$

$= 5 \left(- x {e}^{-} x + {e}^{-} x\right)$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = 5 {e}^{-} x \left(1 - x\right)$.

Alternatively , using Quotient Rule ,

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

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

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = 5 {e}^{-} x \left(1 - x\right)$, as before!

Enjoy Maths.!