homework questions? 2-5: implicit differentiation ©2002 roy l. gover () objectives: define implicit...
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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
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