$$\require{cancel}$$

# 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},

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