Homework questions?

Homework questions?

Homework questions?

2-5: Implicit Differentiation

©2002 Roy L. Gover (www.mrgover.com)

Objectives:•Define implicit and explicit functions

•Learn implicit differentiation

DefinitionA function is in explicit form if the independent variable is stated in terms of the dependent variable. Example:

2y ax bx c

DefinitionA function is in implicit form if the independent variable is not stated in terms of the dependent variable. Example:

2ax bx y c

Important Idea

When a function is in implicit form, it may be difficult to solve for a dependent variable and differentiate explicitly. In this case, implicit differentiation is used.

Example

Differentiate xy=1

Method 1: solve for y and differentiate explicitlyMethod 2: differentiate implicitly…

ExampleDifferentiate xy=1 implicitly. 1d d

xydx dx

1. Use product rule on left.

2. Differentiate x normally;

differentiate y using chain rule.

3. Solve for and simplify. dy

dx

ExampleDifferentiate:

Variables agree-use power rule & differentiate explicitly

42d

xdx

ExampleDifferentiate:

Variables disagree-use power rule & differentiate implicitly

42d

ydx

ExampleFind given that:

dy

dx3 2 22 6 3 5y y y x

Steps:1. Differentiate both sides with respect to x.

ExampleFind given that:

dy

dx3 2 22 6 3 5y y y x

Steps:2. Collect all terms with on left, everything else on right.

dy

dx

ExampleFind given that:

dy

dx3 2 22 6 3 5y y y x

Steps:dy

dx3. Factor out of left side of equation.

ExampleFind given that:

dy

dx3 2 22 6 3 5y y y x

Steps:dy

dx4. Solve for .

3 2 22 6 3 5y y y x

3 2 22 6 3 5y y y x

Try This

Find : dy

dx2 2 3x y y x

Hint: differentiate implicitly using the product rule.

2

2

2

2

dy xy y

dx x xy

Warm-Up

Find and evaluate at (1,-1):

dy

dx2 3 2x y

Fill in the blanks:

The answer is the ______ of the _____ ______at (1,-1).

Try This

2 2 1x y Find the slope, if it exists:

at the points (0,1) and (1,0)at (0,1), slope=0at (1,0), slope is undefined

Try ThisFind the equation of the line tangent to the graph:2 24 4x y at the point

22,

2

12

2y x

Try This

2 2 25x y

Find implicitly:2

2

d y

dx

2

2 3

25d y

dx y

Lesson Close

When would you use implicit differentiation to find the derivative?

Assignment

1. 146/1-15 odd, 35,37,41