15-chain-rule

# 15-chain-rule - Math 1a The Chain Rule Fall 2009 ’& \$...

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Unformatted text preview: Math 1a The Chain Rule Fall, 2009 ’ & \$ % Another Differentiation Rule The Chain Rule: Suppose F ( x ) = f ( g ( x )) (or F = f ◦ g ). Further sup- pose that g is differentiable at x and f is differentiable at g ( x ). Then F is differentiable at x and F ( x ) = f ( g ( x )) g ( x ) or dy dx = dy du du dx (where for the Leibniz notation expression we’re thinking of y = f ( u ) and u = g ( x )). This has consequences for all our previous differentiation techniques: 1 The derivative of y = ( f ( x ) ) n = u n : d dx ( f ( x ) ) n = n ( f ( x ) ) n- 1 · f ( x ) or ( u n ) = nu n- 1 · u (where n is any real number). 2 The derivative of an exponential function: d dx ( e u ) = e u u 3 The derivative of trigonometric functions: d dx sin( u ) = cos( u ) u and d dx cos( u ) =- sin( u ) u . Using these rules, compute the following derivatives. You need not simplify your final answer (except in problems 3 and 4). 1 Find y if y = ( x + 2) 3 . 2 Find dy dx if y = ( x 3 + 2) 3 . 3 Find dx dt if x = t 3 + 1 t 3- 1 . 4 Find dx dt if x = 1 + 2( t 3- 1)- 1 . 5 Find y if y = √ x 3 + x . 6 Find dy dx if y = 1 √ 1 + x 2 7 Find dy dx if y = cos( x 2 ) + sin 2 ( x ). 8 Find y if y = e x 2- e 2 x . 9 Find y if y = x 3 ( x 2 + 1) 4 . 10 Find y if y = 1 ( x 2 + x ) 2 . 11 Find dy dx if y = ( x 2 + x )- 2 . 12 Find y if y = ( x 3 + 1) 3 x 2- x + 1 . 13 Find dy dx if y = 2 x 2 + x 2 2 4 ....
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## This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.

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15-chain-rule - Math 1a The Chain Rule Fall 2009 ’& \$...

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