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### Lesson 6

Course: MATH 1314, Fall 2008
School: U. Houston
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Word Count: 280

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1314 Math 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 h( x) = 3 x 2 5 x + 6 into functions f (x) and g (x) such that h( x) = ( f g )( x). ( ) 4 Rule 10: The Chain Rule d [ f (g (x ))] = f ' (g ( x) )g ' ( x) dx Example 2: Find the derivative if f ( x) = 3 x 3 4 . ( ) 5 Example 3: Find the derivative if f...

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1314 Math 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 h( x) = 3 x 2 5 x + 6 into functions f (x) and g (x) such that h( x) = ( f g )( x). ( ) 4 Rule 10: The Chain Rule d [ f (g (x ))] = f ' (g ( x) )g ' ( x) dx Example 2: Find the derivative if f ( x) = 3 x 3 4 . ( ) 5 Example 3: Find the derivative if f ( x) = 4 x 4 10 . Example 4: Find the derivative if f ( x) = (2 x 5 2 8 ) 4 . We can also apply the chain rule in problems involving the exponential function and the logarithmic function. Rule 11: The Chain Rule (Exponential Function) d f ( x) e = e f ( x ) f ' ( x) dx (Note, we will not work chain rule problems where the exponential function has a base other than e.) [ ] Example 5: Find the derivative: f ( x) = 2e 5 x . Rule 12: Chain The Rule (Logarithmic Function) d [ln | f ( x) |] = f ' ( x) , provided f ( x) > 0 dx f ( x) Example 6: Find the derivative: f ( x) = ln(3 x 2 + 5) . Sometimes it is helpful to use the properties of logarithms to simplify a problem before we find the derivative: Example 7: Find the derivative: f ( x) = ln 5 x 4 ( ) Example 8: Find the derivative: f ( x) = ln x 2 + 1 x 3 5 [( )( 3 )] 4 We can also use the chain rules together with either the product rule or the quotient rule. Example 9: Find the...

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