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 For we get
each term
one at a time
use the chain rule for terms containing y
For we get
Now solve for dy/dx

4. View Spreadsheet Example
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

5. Guidelines for Implicit Differentiation

6. Slope of a Tangent Line
Given x3 + y3 = y + 21 find the slope of the tangent at (3,-2)
3x2 +3y2y’ = y’
Solve for y’
Substitute x = 3, y = -2

7. Second Derivative
Given x2 –y2 = 49
y’ =??
y’’ =
Substitute

8. Exponential & Log Functions
Given y = bx where b > 0, a constant
Given y = logbx
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