1. Implicit Differentiation
Equations hard to solve for y

2. Function defined Implicitly
Definition
Differentiating with Respect to x

3. Implicit Differentiation

4. Implicit Differentiation

5. Guidelines for Implicit Differentiation
Differentiate both sides of the equation with respect to x.
Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation.
Factor dy/dx out of the left side of the equation.
Solve for dy/dx by dividing both sides of the equation by the left-hand factor that does not contain dy/dx.

6. Implicit Differentiation
Representing a Graph by Differentiable Functions
Finding the Slope of a Graph Implicitly
Determining a Differentiable Function
Examples
More examples

7. Finding the Second Derivative
Given

8. Finding a Tangent Line to a Graph
Find equations for the tangent line and the normal line to the curve at the given point.

9. Optical Illusions
Section Project p. 140