Slope of a parametrically-defined curve

Question

A curve has parametric equations

$$x = t − \frac 1t \qquad\qquad\qquad y = 2t + \frac 1t$$

with $t \ne 0$. Show that $$\frac{\mathrm d y}{\mathrm d x} = 2 - \frac{3}{t^2+1}$$

Answer 1

The chain rule tells us that $$\frac{dy}{dt}= \frac{dy}{dx} \frac{dx}{dt} $$

We can easily calculate two of those derivatives, and then solve for the third.

Answer 2

The hint: $$\frac{dy}{dx}=\frac{y'_t}{x'_t}.$$