1. ## Chain rule

Given that y=(1+6x)^1/3 , show that dy/dx=2/y2.

2. ## Re: Chain rule

Please show some work, so we know how to help you on this question. We can only provide hints/ partial solutions, after seeing some work from the poster.

3. ## Re: Chain rule

I end up with 2/y^2/3, and it should be 2/y2.

4. ## Re: Chain rule

$\displaystyle \frac{dy}{dx}=2(1+6x)^{\frac{-2}{3}}=\frac{2}{(1+6x)^{\frac{2}{3}}}=\frac{2}{((1 +6x)^{\frac{1}{3}})^2}$

5. ## Re: Chain rule

There you have it, Shakarri gave you the explicit answer. I suggest that you do more exercises so that these algebraic manipulations become natural for you.

Best of luck.

6. ## Re: Chain rule

Thankyou very much . Yes i need more practice, i also see where i went wrong.

7. ## Re: Chain rule

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*}