1. Implicit differentiation (2.5) October 29th, 2012

2. I. Implicit & explicit functions Def. When differentiating a function y with respect to x we have only been given equations in explicit form (y is explicitly written as a function of x), like. That same equation is defined implicitly by. We can then find using implicit differentiation by using the chain rule to differentiate terms involving y.

3. Ex. 1: Differentiate each expression with respect to x. a. b. c. d.

4. You Try: Differentiate each expression with respect to x. a. b. c. d.

5. II. Implicit differentiation Guidelines for Implicit Differentiation: 1. Differentiate both sides of the equation with respect to x. 2. Collect all terms that contain on the left side of the equation, all the terms without on the right side of the equation. 3. Factor out. 4. Solve for.

6. Ex. 2: Find given that. *To graph, use parametric mode. Since, let and.

7. You Try: Find given that.

8. Ex. 3: If possible, represent y as a differentiable function of x. a. b. c.

9. Ex. 4: Determine the slope of the tangent line to the graph of at the point (2, 1).

10. You Try: Determine the slope of the tangent line to the graph of at the point.

11. Ex. 5: Find implicitly for the equation. Then find the largest interval of the form 0 < y < a on which y is a differentiable function of x.

12. Ex. 6:Given, find in terms of x and y.

13. You Try: Given, find in terms of x and y.

14. Ex. 7: Find the equation of the tangent line to the graph of at the point.

16. You Try: Find the equation of the tangent line to the graph of at.