Given that y=(1+6x)^1/3 , show that dy/dx=2/y^{2}.
Thanks in advance.
If you are expected to get an implicit solution (as you are here) implicit differentiation is usually the most direct method:
$\displaystyle \displaystyle \begin{align*} y &= \left( 1 + 6x \right) ^{\frac{1}{3}} \\ y^3 &= 1 + 6x \\ \frac{d}{dx} \left( y^3 \right) &= \frac{d}{dx} \left( 1 + 6x \right) \\ \frac{d}{dy} \left( y^3 \right) \frac{dy}{dx} &= 6 \\ 3y^2 \,\frac{dy}{dx} &= 6 \\ \frac{dy}{dx} &= \frac{6}{3y^2} \\ \frac{dy}{dx} &= \frac{2}{y^2} \end{align*}$