Implicit Differentiation Objective: To find derivatives of functions that we cannot solve for y.

Implicit Differentiation It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation (but often it is easier to do so). Find dy/dx for . Can we solve this for y?

Implicit Differentiation It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation. For example, we can take the derivative of with the quotient rule:

Implicit Differentiation We can also take the derivative of the given function without solving for y by using a technique called implicit differentiation. We will use all of our previous rules and state the independent variable.

Example 2 Use implicit differentiation to find dy/dx if

Example 2 Use implicit differentiation to find dy/dx if

Example 2 Use implicit differentiation to find dy/dx if

Example 2 Use implicit differentiation to find dy/dx if

Example 3 Use implicit differentiation to find if

Example 3 Use implicit differentiation to find if

Example 3 Use implicit differentiation to find if

Example 3 Use implicit differentiation to find if

Example 3 Use implicit differentiation to find if

Example 3 Use implicit differentiation to find if

Example 3 Use implicit differentiation to find if

Example 4 Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1).

Example 4 Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1). We know that the slope of the tangent line means the value of the derivative at the given points.

Example 4 Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1). We know that the slope of the tangent line means the value of the derivative at the given points.

Example 4 Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1). We know that the slope of the tangent line means the value of the derivative at the given points.

Example 5 Use implicit differentiation to find dy/dx for the equation .

Homework Page 241-242 1-23 odd 27 (just use implicit)