What is implicit differentiation and how do I do it?

Implicit differentiation involves differentiating expressions for which it is difficult to rearrange into y in terms of x. It is important to note which variable is being differentiated by the other as, for example, they may ask for dx/dy rather than dy/dx. In the more common case of dy/dx, terms containing only the x variable can be differentiated normally using the relevant rules. For terms containing y it is useful to separate the term into its x component and y component by labelling the x component u and the y component v. For example 2xy would be split into u = 2x and v = y. You then differentiate u and v separately with respect to x to find du/dx and dv/dx. For the x component this is normal differentiation while for the y term it is easiest to differentiate it in terms of y then multiply by dy/dx for example d(4y^2)/dx = 8ydy/dx. This works due to the chain rule which states dy/dx = dy/dv * dv/dx hence dv/dx = dv/dy * dy/dx. The product rule states dy/dx = vdu/dx + u*dv/dx so sub in each part to find the differential of the term. This can be repeated for each term separately. The equation can then be rearranged to give dy/dx in terms of x and y by taking the terms containing dy/dx to one side of the equation before factorising out dy/dx as the common factor and dividing both sides by the coefficient of dy/dx.