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

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

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},

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