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

Implicit Differentiation MAT 224 SPRING 2007 Implicit Differentiation Example 6 (cont):

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