Section 4.6 Implict differentiation
So far we have mostly dealt by functions that are given “explicitly”, as \(y = f(x)\text{.}\) But sometimes functions are defined “implicitly”: this happens when a function \(y = f(x)\) is defined implicitly by a relation between \(y\) and \(x\) (which you may or may not be able to solve explicitly for \(y\)). We study such implicit functions in this section, and show how we can calculate their derivatives. The resulting process is known as “implicit differentiation”.
Objectives
You should be able to:
- Explain and illustrate the concept of implicit functions.
- Use the method of implicit differentiation to calculate the derivative of implicit functions.
Subsection 4.6.1 Instructional video
Subsection 4.6.2 Key concepts
Concept 4.6.1. Implicit functions.
Given a relation
we say that \(f\) is a function defined implicitly by this relation if
for all \(x\) in the domain of \(f\text{.}\)
A relation \(H(x,y)=0\) defines a curve in the \(xy\)-plane, which implicitly defines \(y\) as one or several functions of \(x\text{.}\)
Concept 4.6.2. Implicit differentiation.
Implicit differentiation is the process of calculating the derivative of a function defined implicitly by a relation \(H(x,y)=0\text{.}\)
To calculate the derivative \(y'\) of an implicit function:
- Treat the variable \(y\) in the relation as an unknown but differentiable function of \(x\) (like \(y=g(x)\)), and differentiate both sides of the relation with respect to \(x\text{,}\) using the chain rule.
- Collect the terms involving \(y'\) on one side of the equation and solve for \(y'\text{.}\)
Note that this will generally give \(y'\) as a function of \(x\) and \(y\text{,}\) where \(y\) is understood as the function implicitly defined by the original relation.