Implicit Differentiation

1
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 cant 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
Simple power rule
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Implicit Differentiation
If variables do not agree, then use the chain
rule.
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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
Example 6 Find the points at which the graph of
the equation has a horizontal tangent line.
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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