1. 1 Implicit Differentiation

2. 2 Introduction Consider an equation involving both x and y: This equation implicitly defines a function in x It could be defined explicitly

3. 3 Differentiate Differentiate both sides of the equation –each term –one at a time –use the chain rule for terms containing y For we get Now solve for dy/dx

4. 4 Differentiate Then gives us We can replace the y in the results with the explicit value of y as needed This gives us the slope on the curve for any legal value of x

5. 5 Guidelines for Implicit Differentiation

6. 6 Slope of a Tangent Line Given x 3 + y 3 = y + 21 find the slope of the tangent at (3,-2) 3x 2 +3y 2 y’ = y’ Solve for y’ Substitute x = 3, y = -2

7. 7 Second Derivative Given x 2 –y 2 = 49 y’ =?? y’’ = Substitute

8. 8 Find the derivative with respect to x.

9. 9 1. Find the equation of the tangent line to at the point (2,1).

10. 10 2.Find the second derivative with respect to x. Look for a substitution of the original.

11. 11 3.Find the points at which the graph has horizontal and vertical tangents. Horizontal: Vertical:

12. 12 We need the slope. Since we can’t solve for y, we use implicit differentiation to solve for. Find the equations of the lines tangent and normal to the curve at. Note product rule.

13. 13 Find the equations of the lines tangent and normal to the curve at. tangent:normal:

14. 14 Higher Order Derivatives Find if. Substitute back into the equation.