Implicit differentiation

If we want to differentiate an implicit function such as , we notice that by the chain rule,

$$\frac{\mathrm{d}}{\mathrm{d}x} f(y) = f'(y) \frac{\mathrm{d}y}{\mathrm{d}x}$$

We can use this results to differentiate the above function as follows.

$$\begin{eqnarray*} y^2 + y & = & x + x^3 \\ 2y \frac{\mathrm{d}y}{\mathrm{d}x} +... ... \frac{\mathrm{d}y}{\mathrm{d}x} & = & \frac{1 + 3 x^2}{2y + 1} \end{eqnarray*}$$