# What is the derivative of f(t) = (t/(t+2) , t/(t^2-1) ) ?

Jul 20, 2018

$f ' \left(t\right) = \left(\frac{2}{t + 2} ^ 2 , - \frac{{t}^{2} + 1}{{t}^{2} - 1} ^ 2\right)$

#### Explanation:

We have to keep this in parametric form, so we use the quotient rule and find
$\frac{d}{\mathrm{dt}} \left(f \left(t\right)\right) = \left(\frac{d}{\mathrm{dt}} \left(\frac{t}{t + 2}\right) , \frac{d}{\mathrm{dt}} \left(\frac{t}{{t}^{2} - 1}\right)\right)$
$= \left(\frac{\left(t + 2\right) \cdot 1 - t \cdot 1}{{\left(t + 2\right)}^{2}} , \frac{\left({t}^{2} - 1\right) - t \cdot \left(2 t\right)}{{\left({t}^{2} - 1\right)}^{2}}\right)$
$= \left(\frac{2}{t + 2} ^ 2 , - \frac{{t}^{2} + 1}{{t}^{2} - 1} ^ 2\right)$