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

Jul 22, 2017

Derivative of $f \left(t\right)$ is $- \frac{2 t}{\left(2 t - \cos t\right) {\left({t}^{2} - 1\right)}^{2}}$

#### Explanation:

The derivative of a function $f \left(t\right) = \left(x \left(t\right) , y \left(t\right)\right)$ is given by $\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}$

as $x \left(t\right) = {t}^{2} - \sin t$ we have $\frac{\mathrm{dx}}{\mathrm{dt}} = 2 t - \cos t$ and

for $y \left(t\right) = \frac{1}{{t}^{2} - 1}$ we have $\frac{\mathrm{dy}}{\mathrm{dt}} = - \frac{2 t}{{t}^{2} - 1} ^ 2$

Hence derivative of $f \left(t\right) = \left({t}^{2} - \sin t , \frac{1}{{t}^{2} - 1}\right)$

$\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}} = \frac{- \frac{2 t}{{t}^{2} - 1} ^ 2}{2 t - \cos t} = - \frac{2 t}{\left(2 t - \cos t\right) {\left({t}^{2} - 1\right)}^{2}}$