# How do you differentiate the following parametric equation:  x(t)=-2te^t+4t, y(t)= -5t^2-e^(t) ?

Oct 29, 2016

$\frac{\mathrm{dx}}{\mathrm{dt}} = 4 - 2 \left(1 + t\right) {e}^{t}$
$\frac{\mathrm{dy}}{\mathrm{dt}} = - 10 t - {e}^{t}$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{4 - 2 \left(1 + t\right) {e}^{t}}{- 10 t - {e}^{t}}$

#### Explanation:

Given:
$x \left(t\right) = 4 t - 2 t {e}^{t}$
$y \left(t\right) = - 5 {t}^{2} - {e}^{t}$

Differentiate both:

$\frac{\mathrm{dx}}{\mathrm{dt}} = 4 - 2 \left(1 + t\right) {e}^{t}$
$\frac{\mathrm{dy}}{\mathrm{dt}} = - 10 t - {e}^{t}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{4 - 2 \left(1 + t\right) {e}^{t}}{- 10 t - {e}^{t}}$
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