# How do you find the derivative of Y= ( 5x + 3 )( x - 7)^(-1/3)?

Jun 3, 2015

In this way with the product rule:

$y ' = 5 {\left(x - 7\right)}^{- \frac{1}{3}} + \left(5 x + 3\right) \left(- \frac{1}{3}\right) {\left(x - 7\right)}^{- \frac{1}{3} - 1} =$

$= \frac{1}{3} {\left(x - 7\right)}^{- \frac{4}{3}} \cdot \left[3 \cdot 5 \cdot {\left(x - 7\right)}^{- \frac{1}{3} - \left(- \frac{4}{3}\right)} - \left(5 x + 3\right) \cdot 1\right] =$

$= \frac{1}{3} {\left(x - 7\right)}^{- \frac{4}{3}} \cdot \left[15 \cdot \left(x - 7\right) - 5 x - 3\right] =$

$= \frac{1}{3} {\left(x - 7\right)}^{- \frac{4}{3}} \left(10 x + 102\right) =$

$= \frac{2}{3} {\left(x - 7\right)}^{- \frac{4}{3}} \left(5 x + 51\right)$.