# How do you differentiate the following parametric equation:  x(t)=lnt, y(t)= tcost ?

Jan 4, 2016

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

#### Explanation:

$x ' \left(t\right) = \frac{1}{t}$

Product rule here:

$y ' \left(t\right) = \cos t - t \sin t$

The derivative of a parametric function can be found through:

$\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}} = \frac{y ' \left(t\right)}{x ' \left(t\right)} = \frac{\cos t - t \sin t}{\frac{1}{t}} = t \left(\cos t - t \sin t\right)$