Use logarithmic differentiation to determine y' for the equation y = V(10x – 3)(x – 4)(2x + 5). Write your answer in terms of x only.

Given , $y=\sqrt[3]{\left(10x-3\right)\left(x-4\right)\left(2x+5\right)}$ 

           $y=\left(\left(10x-3\right)\left(x-4\right){\left(2x+5\right)}^{\frac{1}{3}}\right)$

Taking log on both sides for $\text{'}y\text{'}$

$\log \left(y\right)=\log \left(\left(10x-3\right)\left(x-4\right){\left(2x+5\right)}^{\frac{1}{3}}\right)$

$\log \left(y\right)=\frac{1}{3}\left\{\log \left(\left(10x-3\right)\left(x-4\right)\left(2x+5\right)\right)\right\}$

On differentiating the above equation,

$\frac{d}{dx}\left(\log \left(y\right)\right)=\frac{d}{dx}\left\{\frac{1}{3}\left[\log \left(\left(10x-3\right)\left(x-4\right)\left(2x+5\right)\right)\right]\right\}$