Lect15 - Product Rule Unfortunately the derivative of a...

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92.131 Lecture 15 1 of 8 Ronald Brent © 2008 All rights reserved. Product Rule Unfortunately the derivative of a product is not the product of the derivatives. For example, one can write ) ( ) ( ) ( 2 x v x u x x p = = where x x v x u = = ) ( ) ( . x x p 2 ) ( = and 1 1 1 ) ( ) ( = = x v x u , and they are not equal in general. To compute the derivative of a product consider ) ( ) ( ) ( x v x u x p = , where the two functions u and v are both differentiable. Using the limit definition: h x v x u h x v h x u h x p h x p x p h h ) ( ) ( ) ( ) ( lim ) ( ) ( lim ) ( 0 0 + + = + = By subtracting and adding ) ( ) ( h x v x u + we can write h x v x u h x v x u h x v x u h x v h x u x p h ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( lim ) ( 0 + + + + + =
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92.131 Lecture 15 2 of 8 Ronald Brent © 2008 All rights reserved. Which can be rewritten as ( )( ) h x v h x v x u x u h x u h x v x p h ) ( ) ( ) ( ) ( ) ( ) ( lim ) ( 0 + + + + = ( ) () h x v h x v x u h x u h x u h x v h h ) ( ) ( ) ( lim ) ( ) ( ) ( lim 0 0 + + + + = ( ) ( ) h x v h x v x u h x u h x u h x v h h h ) ( ) ( lim ) ( ) ( ) ( lim
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Lect15 - Product Rule Unfortunately the derivative of a...

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