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

Aug 8, 2016

$= - t = - \frac{1}{x}$

#### Explanation:

assuming youre looking for $\frac{\mathrm{dy}}{\mathrm{dx}}$, you use the following fact

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}$

$= \frac{\frac{1}{t}}{- \frac{1}{t} ^ 2}$

$= - t$

and if you like

$= - \frac{1}{x}$ as $x = \frac{1}{t}$

we can check this by de-parameterising the equation

from $t = \frac{1}{x}$ we have $y = \ln \left(\frac{1}{x}\right) = - \ln x$ so $\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{x}$