The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages. 
Home Differentiation Implicit Functions Implicit Functions  
Search the VIAS Library  Index  
Implicit Functions
We now turn to the topic of implicit differentiation. We say that y is an implicit function of x if we are given an equation σ(x, y) = τ(x, y) which determines y as a function of x. An example is x + xy = 2y. Implicit differentiation is a way of finding the derivative of y without actually solving for y as a function of x. Assume that dy/dx exists. The method has two steps:
In Example 1, we found dy/dx by three different methods. (a) Implicit differentiation. We get dy/dx in terms of both x and y. (b) Solve for y as a function of x and differentiate directly. This gives dy/dx in terms of x only. (c) Solve for x as a function of y, find dx/dy directly, and use the Inverse Function Rule. This method gives dy/dx in terms of y only.


Home Differentiation Implicit Functions Implicit Functions 