# What is the derivative of f(t) = (1/(3t) , t/(5t-4) ) ?

May 22, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{\left(5 t - 4\right)}^{2}}{12 {t}^{2}}$

#### Explanation:

To find the derivative, use the formula:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(\mathrm{dy} \text{/"dt)/(dx"/} \mathrm{dt}\right)$

First, find $\frac{\mathrm{dy}}{\mathrm{dt}}$:

$\frac{\mathrm{dy}}{\mathrm{dt}} = \frac{d}{\mathrm{dt}} \left(\frac{1}{3 t}\right) = - \frac{1}{3 {t}^{2}}$

Next, find $\frac{\mathrm{dx}}{\mathrm{dt}}$:

$\frac{\mathrm{dx}}{\mathrm{dt}} = \frac{d}{\mathrm{dt}} \left(\frac{t}{5 t - 4}\right) = \frac{\left(5 t - 4\right) - 5 t}{5 t - 4} ^ 2 = - \frac{4}{5 t - 4} ^ 2$

Now, use the formula to find $\frac{\mathrm{dy}}{\mathrm{dx}}$:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(\mathrm{dy} \text{/"dt)/(dx"/} \mathrm{dt}\right) = \frac{- \frac{1}{3 {t}^{2}}}{- \frac{4}{5 t - 4} ^ 2} = \frac{{\left(5 t - 4\right)}^{2}}{12 {t}^{2}}$