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

Dec 11, 2017

Derivative is $\frac{{e}^{t} - \sin t - t \cos t}{{e}^{t} + \sin t - 1}$

#### Explanation:

The differential of a parametric equation of the type $f \left(x \left(t\right) , y \left(t\right)\right)$ is given by $\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}$.

Here $x \left(t\right) = {e}^{t} - \cos t - t$ hence $\frac{\mathrm{dx}}{\mathrm{dt}} = {e}^{t} + \sin t - 1$

and as $y \left(t\right) = {e}^{t} - t \sin t$, $\frac{\mathrm{dy}}{\mathrm{dt}} = {e}^{t} - \sin t - t \cos t$

Hence $\frac{\mathrm{df}}{\mathrm{dx}} = \frac{{e}^{t} - \sin t - t \cos t}{{e}^{t} + \sin t - 1}$