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2 -5: Implicit Differentiation Objectives: • Define implicit and explicit functions • Learn implicit

Definition A function is in explicit form if the independent variable is stated in

Definition A function is in implicit form if the independent variable is not stated

Important Idea When a function is in implicit form, it may be difficult to

Example Differentiate xy=1 Method 1: solve for y and differentiate explicitly Method 2: differentiate

Example Differentiate xy=1 implicitly. 1. Use product rule on left. 2. Differentiate x normally;

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

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

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

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

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 1): and evaluate at (1, - Fill in the blanks: The answer

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

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

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

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Homework questions?

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 Steps: given that: 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 1): and evaluate at (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. 146/1 -15 odd, 35, 37, 41