# How do you differentiate the following parametric equation:  x(t)=e^t-1/t, y(t)=1-t^2 ?

Hi there!

Anytime you're given parametric equations to differentiate, you use the relationship whereby:

dy/(dx) = (dy/(dt))/(dx/(dt)

#### Explanation:

Let's start off by differentiating each equation separately, then combine the two afterwards!

Differentiating y with respect to t (dy/dt) we get:

$\frac{\mathrm{dy}}{\mathrm{dt}} = - 2 t$

Differentiating x with respect to t (dx/dt) we get:

$\frac{\mathrm{dx}}{\mathrm{dt}} = {e}^{t} - \frac{1}{t} ^ 2$

(If you're not sure how I got the $- \frac{1}{t} ^ 2$ from $\frac{1}{t}$... If you convert the $\frac{1}{t}$ to ${t}^{-} 1$ and differentiate using the power rule, it comes out to $- {t}^{-} 2$ which is equivalent to $- \frac{1}{t} ^ 2$)

Now putting everything together we get:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- 2 t}{{e}^{t} - \frac{1}{t} ^ 2}$