1 Implicit Differentiation Lesson 3.5

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

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 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 View Spreadsheet Example View Spreadsheet Example

5 Guidelines for Implicit Differentiation

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

8 Exponential & Log Functions Given y = b x where b > 0, a constant Given y = log b x Note: this is a constant

9 Using Logarithmic Differentiation Given Take the log of both sides, simplify Now differentiate both sides with respect to x, solve for dy/dx

10 Implicit Differentiation on the TI Calculator On older TI calculators, you can declare a function which will do implicit differentiation: Usage: Newer TI’s already have this function

11 Assignment Lesson 3.5 Page 171 Exercises 1 – 81 EOO