# How do you find dy/dx for the curve x=t*sin(t), y=t^2+2 ?

Aug 28, 2014

To find the derivative of a parametric function, you use the formula:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}$, which is a rearranged form of the chain rule.

To use this, we must first derive $y$ and $x$ separately, then place the result of $\frac{\mathrm{dy}}{\mathrm{dt}}$over $\frac{\mathrm{dx}}{\mathrm{dt}}$.

$y = {t}^{2} + 2$

$\frac{\mathrm{dy}}{\mathrm{dt}} = 2 t$ (Power Rule)

$x = t \sin \left(t\right)$

$\frac{\mathrm{dx}}{\mathrm{dt}} = \sin \left(t\right) + t \cos \left(t\right)$ (Product Rule)

Placing these into our formula for the derivative of parametric equations, we have:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}} = \frac{2 t}{\sin \left(t\right) + t \cos \left(t\right)}$