1 Implicit Differentitation
The technique of implicit differentiation allows you to find the derivative of y with respect to x without having to solve the given equation for y. The chain rule must be used whenever the function y is being differentiated because of our assumption that y may be expressed as a function of x.
1.1 Theorem 3.5.1
Suppose that \(I\) is an open interval, and for every number \(x\) in the interval \(I\), \(f(x) = g(x)\). Then for every number \(x\) in \(I\), \(f'(x) = g'(x)\), where we interepret this equation to mean that either both derivatives are defined and they are equal, or both are undefined.