Lesson 6-filled - ) ( 2 + = x x f . Sometimes it is helpful...

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Math 1314 Lesson 6 The Chain Rule In this lesson, you will learn the last of the basic rules for finding derivatives, the chain rule. Example 1 : Decompose ( 29 4 2 6 5 3 ) ( + - = x x x h into functions ) ( x f and ) ( x g such that ). )( ( ) ( x g f x h r = Rule 10: The Chain Rule ( 29 ( 29 [ ] ( 29 ) ( ' ) ( ' x g x g f x g f dx d =
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Example 2 : Find the derivative if ( 29 5 3 4 3 ) ( - = x x f . Example 3 : Find the derivative if 10 4 ) ( 4 - = x x f .
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Example 4 : Find the derivative if ( 29 4 2 8 2 5 ) ( - = x x f . We can also apply the chain rule in problems involving the exponential function and the logarithmic function. Rule 11: The Chain Rule (Exponential Function) [ ] ) ( ' ) ( ) ( x f e e dx d x f x f = (Note, we will not work chain rule problems where the exponential function has a base other than e .) Example 5 : Find the derivative: x e x f 5 2 ) ( = .
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Rule 12: The Chain Rule (Logarithmic Function) [ ] ) ( ) ( ' | ) ( | ln x f x f x f dx d = , provided 0 ) ( x f Example 6 : Find the derivative: ) 5 3 ln(
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Unformatted text preview: ) ( 2 + = x x f . Sometimes it is helpful to use the properties of logarithms to simplify a problem before we find the derivative: Example 7 : Find the derivative: ( 29 4 5 ln ) ( x x f = Example 8 : Find the derivative: ( 29 ( 29 [ ] 4 3 3 2 5 1 ln ) (-+ = x x x f We can also use the chain rules together with either the product rule or the quotient rule. Example 9: Find the derivative: x e x x f 4 2 ) ( = Example 10 : Find the derivative if ( 29 4 3 2 5 2 ) (-= x x x f . From this lesson, you should be able to Apply the chain rules to appropriate problems to find derivatives Use the chain rule, together with other rules, to find derivatives Use log properties to simplify log problems before finding derivatives...
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Lesson 6-filled - ) ( 2 + = x x f . Sometimes it is helpful...

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