2.5 Implicit Differentiation

Implicit and Explicit Functions Explicit FunctionImplicit Function But what if you have a function like this…. To differentiate: Use implicit differentiation.

Implicit Differentiation To find dy/dx implicitly, you must realize that the differentiation is taking place with respect to x. x alone  derive as usual y?  need to use chain rule!

Examples: Differentiating with Respect to x Variables agree  use simple power rule Variables disagree  use chain rule

Examples: Differentiating with Respect to x

Guidelines for Implicit Differentiation 1.Differentiate both sides of the equation with respect to x. 2.Collect all terms involving dy/dx on the left side and all other terms to the right. 3.Factor out dy/dx. 4.Solve for dy/dx

This is not a function, but it would still be nice to be able to find the slope. Example 1:

This can’t be solved for y. Example 2:

Find the equation of the tangent line to the curve at (-1, 2) Example 3:

Higher Order Derivatives Find if. Example 4:

More Examples