 Implicit Differentiation

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Implicit Differentiation
Section 2.5

Explicit Differentiation
You have been taught to differentiate functions in explicit form, meaning y is defined in terms of x. Examples: The derivative is Whenever you can solve for y in terms of x, do so.

Explicit Differentiation
Example: Find Whenever possible, rewrite in explicit form (solve for y). Then take the derivative of y with respect to x.

Implicit Differentiation
Sometimes, however, y can’t be written in terms of x as demonstrated in the following: We need to differentiate implicitly.

Implicit Differentiation
Remember, we are differentiating with respect to x. Using the general power rule and chain rule, we have Variables agree Simple power rule

Implicit Differentiation
If variables do not agree, then use the chain rule. Variables disagree Variables disagree Variables disagree

Implicit Differentiation
Using Implicit Differentiation to Find dy/dx: Four Steps to Success Differentiate both sides of the equation with respect to x. Get all terms containing dy/dx alone on one side of the equation. Factor out dy/dx. Solve for dy/dx by dividing both sides of the equation by the expression remaining in parentheses.

Implicit Differentiation
Example 1:

Implicit Differentiation
Example 2:

Implicit Differentiation
Example 3: Determine the slope of the tangent line to the graph of at the point

Implicit Differentiation
Example 4: Determine the slope of the graph of at the point (-1, 1).

Implicit Differentiation
Example 5: Find the equation of the tangent line of the graph at (-1,2).

Implicit Differentiation
MAT SPRING 2007 Implicit Differentiation Example 6: Find the points at which the graph of the equation has a horizontal tangent line.

Implicit Differentiation
MAT SPRING 2007 Implicit Differentiation Example 6 (cont):

Homework Section 2.5 page 146 #1, 5, 7, 11, 21, 25, 27, 29, 31, 59