Use implicit differentiation to find the derivative of 2yx^2, with respect to x.

To solve this question we use the product rule, where we differentiate one variable whilst keeping the other constant, and vice versa, adding the two results together to get our answer. A helpful formula is (dy/dx)=u(dv/dx)+v(du/dx), which is easy to remember as it is very catchy! To start, we need to choose u and v. You can make the solution easier to get to if you choose suitable u's and v's (Although you will still arrive at the same answer regardless), so here the best option personally would be making u=y, and v=2x^2. As the formula suggests, we need to find (dv/dx) and (du/dx), so we start off by differentiating our v with respect to x. This obtains (dv/dx)=4x. Next, we find (du/dx), but this is slightly more difficult as we have to differentiate y with respect to x. The way I like to think about this is if you were differentiating something simple such as y=x. You would write (dy/dx)=1 without hesistating, but without knowing it you've differentiated y with respect to x here, and this is the same in this case of u=y, where differentiating obtains (du/dx)=(dy/dx). Now we have all the parts to our equation, and now we just sub them in. After subbing u, v, (du/dx) and (dv/dx) into the equation, we arrive at our answer of: (dy/dx)=4xy+2(x^2)(dy/dx) I find after a lot of practice, you will learn not only how to choose your u and v as effectively as possible, but also how to do it in your head!