2.3: Implicit Differentiation

Equation 2.2.5 can be used to solve the problem of differentiation of an implicit function. Consider, for example, the unlikely equation

$$\ln ( xy) = x^2 y^3 \label{2.3.1}$$

Calculate the derivative dy/dx. It would be easy if only one could write this in the form y = something; but it is difficult (impossible as far as I know) to write y explicitly as a function of $x$. Equation $\ref{2.3.1}$ implicitly relates $y$ to $x$. How are we going to calculate $dy/dx$?

The curve $f(x, y) = 0$ might be considered as being the intersection of the surface $z = f (x , y)$ with the plane $z = 0$. Seen thus, the derivative $dy/dx$ can be thought of as the limit as $δx$ and $δy$ approach zero of the ratio $δy/δx$ within the plane $z = 0$; that is, keeping z constant and hence $δz$ equal to zero. Thus equation 2.2.5 gives us that

$$\frac{dy}{dx} = - \left( \frac{\partial f}{\partial x} \right) / \left( \frac{\partial f}{\partial y} \right).$$

For example, show that, for Rquation $\ref{2.3.1}$,

$$\frac{dy}{dx} = \frac{y(2x^2y^3-1)}{x(1-3x^2y^3)}.$$