# 2.3: Implicit Differentiation

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

\[ \ln ( xy) = x^2 y^3\]

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 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 δ*x* and δ*y* approach zero of the ratio δy/δx within the plane *z* = 0; that is, keeping *z* constant and hence δ*z* equal to zero. Thus equation 2.2.5 gives us that

\[ \frac{dy}{dx} = - \left( \frac{ \delta f}{ \delta x} \right) / \left( \frac{ \delta f}{ \delta y} \right).\]

For example, show that, for equation 2.3.1,

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