Product rule does come into it with the first part of the equation but you need to apply the chain rule as well.
first of all when you differentiate y it should become
So applying the chain rule to you get:
you deal with in a similar way.
Applying all this with the product rule the whole function should become:
You now need to rearrange to get
Put the on the other side
Take out as a common factor
Then divide to leave on its own
Differentiating y can be confusing so I can try to go over it in more detail if you want.
3xy^2 + y^3 = 8
now product rule...
(3)(y^2)dx/dy + (3x)(2y) dx/dy +3y^2 dx/dy = 0
I have all the dx/dy on one side... now what? This is where I went wrong and I do know know where...
Oh ok thanks alot, but when do you know to add the dy/dx beside one of the terms? I think that is where I am going wrong.
Notice on the LHS you have a sum, and the derivative of a sum is the same as the sum of the derivatives. On the right you have a constant, and the derivative of a constant is 0.
So we have
The first part is a product, so we need to use the product rule. Since y is a function of x, we can find a derivative with respect to x, in terms of y. We do this using the chain rule.
Notice that . Also note that we are trying to FIND .
So, using these pieces of information we have
Hope that helped. Have a go of the second one yourself.