Implicit Differentiation Lesson 3.5

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

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

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

Guidelines for Implicit Differentiation

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

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

Exponential & Log Functions Given y = bx where b > 0, a constant Given y = logbx Note: this is a constant

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

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

Assignment Lesson 3.5 Page 171 Exercises 1 – 81 EOO