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2-5: Implicit Differentiation ©2002 Roy L. Gover ( Objectives: Define implicit and explicit functions Learn implicit differentiation

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

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

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 explicitly Method 2: differentiate implicitly…

Example Differentiate xy =1 implicitly. 1. Use product rule on left. 2. Differentiate x normally; differentiate y using chain rule. 3. Solve for and simplify.

Example Differentiate: Variables agree-use power rule & differentiate explicitly

Example Differentiate: Variables disagree-use power rule & differentiate implicitly

Example Find given that: Steps: 1. Differentiate both sides with respect to x.

Example Find given that: Steps: 2. Collect all terms with on left, everything else on right.

Example Find given that: Steps: 3. Factor out of left side of equation.

Example Find given that: Steps: 4. Solve for.

Try This Find : Hint: differentiate implicitly using the product rule.

Warm-Up Find and evaluate at (1,-1): Fill in the blanks: The answer is the ______ of the _____ ______at (1,-1).

Try This Find the slope, if it exists: at the points (0,1) and (1,0) at (0,1), slope=0 at (1,0), slope is undefined

Try This Find the equation of the line tangent to the graph: at the point

Try This Find implicitly:

Lesson Close When would you use implicit differentiation to find the derivative?

Assignment /1-15 odd, 35,37,41