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**Implicit Differentiation**

Section 2.5

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**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.

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**Explicit Differentiation**

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

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**Implicit Differentiation**

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

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**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

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**Implicit Differentiation**

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

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**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.

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**Implicit Differentiation**

Example 1:

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**Implicit Differentiation**

Example 2:

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**Implicit Differentiation**

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

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**Implicit Differentiation**

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

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**Implicit Differentiation**

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

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**Implicit Differentiation**

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

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**Implicit Differentiation**

MAT SPRING 2007 Implicit Differentiation Example 6 (cont):

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

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