notes-chainrule

# 2 ² 4 2 x 1 ² 1 2 x 1 2 ³ simplify 2 x 1 2 ² 8 x

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2 ² 4) + (2 x + 1) ² 1 2 x ° 1 = 2 ³ (simplify) = 2 x 1 = 2 ² 8 + x 1 = 2 + 1 2 x ° 1 = 2 = x 1 = 2 + 1 2 x ° 1 = 2 ² 8 Now put it all together y 0 = f 0 ± g 0 = 3 h (2 x + 1)( x 1 = 2 ² 4) i 2 ² x 1 = 2 + 1 2 x ° 1 = 2 ² 8 ³ 2. Example : suppose y = ´ 6 x + 1 x 3 µ 2 Here this one is a little more complicated. The outside function ( f ) is f = ( ) 2 but the nested function ( g ) is g ( x ) = 6 x + 1 x 3 3

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.which will require the quotient rule rule. This one should be done in parts seperately then subsitute into equation 1. First g 0 = (6)( x 3 ) ² (2 x + 1)(3 x 2 ) ( x 3 ) 2 = 6 x 3 ² 6 x 3 ² 3 x 2 x 6 = ² 3 x 2 x 6 = ² 3 x ° 4 remember that f = ( ) 2 and f 0 = 2 ( ) ; we can use equation 1 y 0 = f 0 [ g ( x )] ° g 0 ( x ) = 2 f 0 z }| { ´ 6 x + 1 x 3 µ ± g 0 z }| { ° ² 3 x ° 4 ± 4
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