Section 4.6 Implict 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
H(x,y)=0,H(x,y)=0,
we say that ff is a function defined implicitly by this relation if
H(x,f(x))=0H(x,f(x))=0
for all xx in the domain of f.f.
A relation H(x,y)=0H(x,y)=0 defines a curve in the xyxy-plane, which implicitly defines yy as one or several functions of x.x.
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.H(x,y)=0.
To calculate the derivative y′y′ of an implicit function:
- Treat the variable yy in the relation as an unknown but differentiable function of xx (like y=g(x)y=g(x)), and differentiate both sides of the relation with respect to x,x, using the chain rule.
- Collect the terms involving y′y′ on one side of the equation and solve for y′.y′.
Note that this will generally give y′y′ as a function of xx and y,y, where yy is understood as the function implicitly defined by the original relation.
Further readings 4.6.3 Further readings
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