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

Feb 12, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{2 t \sin t} + \frac{1}{2 {t}^{2} \cos t}$

#### Explanation:

Given that the equations are
$x \left(t\right) = {\cos}^{2} t$, $y \left(t\right) = \sin \frac{t}{t}$

The derivative of a parametric equation when in the form $\frac{\mathrm{dy}}{\mathrm{dx}}$ can be re-written as $\frac{\mathrm{dy} / \mathrm{dt}}{\mathrm{dx} / \mathrm{dt}}$

For $y \left(t\right)$ using quotient rule,
$\frac{\mathrm{dy}}{\mathrm{dt}} = \frac{t \cos t - \sin t}{t} ^ 2$

Similarly, for $x \left(t\right)$, using product rule,
$\frac{\mathrm{dx}}{\mathrm{dt}} = - 2 \cos t \sin t$

So, the parametric differentiative turns to
$\frac{\left(t \cos t - \sin t\right) / {t}^{2}}{- 2 \sin t \cos t}$

Simplifying it shouldn't be a hassle to simplify.