# How do you differentiate the following parametric equation:  x(t)=t-te^t , y(t)=e^t ?

Oct 19, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = {e}^{t} / \left(1 - {e}^{t} - t {e}^{t}\right)$

#### Explanation:

Start dy differentiating x and y with respect to t
$\frac{\mathrm{dx}}{\mathrm{dt}} = 1 - \left({e}^{t} + t {e}^{t}\right) = 1 - {e}^{t} - t {e}^{t}$
and $\frac{\mathrm{dy}}{\mathrm{dt}} = {e}^{t}$
So $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}} = {e}^{t} / \left(1 - {e}^{t} - t {e}^{t}\right)$