lecture20 - u = g ( x ), then y = m ( x ) is a composite...

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Math 006 (Lecture 20) General Power Rule Example 1. Find the derivative of f ( x ) = ( x + 1) 2 . Theorem 1. (General Power Rule) If u ( x ) is a differentiable function and n is any real number, and y = f ( x ) = [ u ( x )] n . Then, dy dx = f 0 ( x ) = n [ u ( x )] n - 1 u 0 ( x ) . Example 2. Find the derivative for each of the following functions: (a) (3 x + 1) 4 (b) ( x 3 + 4) 11 (c) 1 ( t 2 + t + 1) 5 (d) x 2 - 1 1
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Example 3. Find the equation of the line tangent to the graph of y = f ( x ) at x = 2 for f ( x ) = x 2 x + 12 Example 4. Find the derivative for each of the following functions: (a) 2 x x 2 + 1 (b) p (3 x - 1) 3 ( x 2 + 1) 1 / 3 (c) r 3 x + 1 2 x 2 - 1 Example 5. Find the value(s) of x where the tangent line of f ( x ) = 4 x 2 - 2 x is horizontal. 2
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Chain Rule Definition 1. If y = f ( u ) and
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Unformatted text preview: u = g ( x ), then y = m ( x ) is a composite function if y = m ( x ) = f ( g ( x )) Example 6. Find y = m ( x ) = f ( g ( x )) if (a) f ( u ) = u 5 , g ( x ) = 2 x +1 (b) f ( u ) = 2 u +1 , g ( x ) = x 5 (c) f ( u ) = e u , g ( x ) = x 2 + 1 Theorem 2. (Chain Rule) Suppose y = m ( x ) = f ( g ( x )) . Then m ( x ) = f ( g ( x )) g ( x ) . Example 7. Find the derivative for each of the following functions: (a) e x 2 (b) ln(1 + 2 x 4 ) (c) √ 1 + 2 e x 2 +1 (d) x ln(4 x 6 + x-1) (e) x + 1 2 + e 3 x 3...
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This note was uploaded on 09/04/2010 for the course MATH MATH006 taught by Professor Forgot during the Fall '09 term at HKUST.

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lecture20 - u = g ( x ), then y = m ( x ) is a composite...

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