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Question 1)

\f(1 point) Let f be function such that f(1) = 2 and f'(1) = 9. Find h'(1) for the function h(x) = f(x) f(x). h' (1) =(1 point) Calculate g'(x), where g( ) is the inverse of f(I) = g'(x) =c-1 (1 point) Let f(x) = (a) Evaluate f ' (3) = (b) Evaluate (f )'(3) =(2x - 1 ) - x ( ex - 1 )2 (x - 1 1 2 quotient rule . to ( y ) Y = V 2 8 . 4 Given function . f ( ) = 2-1 2+ 1 Change f( x ) to y . Y= 2 - 1 *+ 1 Swap roles of x & y, 2 = 1 + 1 Now , solve for y ( The inverse), xy+x = nal x y - y = - 1 -x = ) y - xy = x +1 > ) 7 ( 1- x ) = 2+1 2 + 1 1 - x " fol ( * ) = ! Now , a ) 1 -1 (3 ) = 3+1 -+ - 2 1 - 3 6 ) ( f -1 ( 24 ) ] =. (1-x ) 2 ( *+1 ) - (x +1 ) the ( 1 - x ) 1- X = (1 - x 12 = ( 1- X ) + ( x +1 ) ( 1 - x ]2 2 ( 1 - x ) ". ( f - 1 ) ' ( 3 ) = 2 ( 1 - 3 ) 2Given, f ( x ) = 2 Change foxy to y . ' 2+1 Swap roles of my y. Now, solve for y ( The inverse ) xy + x = y xy - y = - x 2 4 ( 2 - 1) = - 2 7 = X - 1 f-1 ( x ) = - 5 X - 1. : 9 (x ) is the inverse of fool . 7 ( 2 ) = fri ( 2 ) = - - (x - 1 ) . am ( 2 ) - x x ( x -4 ) ( x - 1 ) 28 .1 . Given function, of ( x ) = 3 / x 1/3 . .. f ' ( x ) = TX ( 2 - /3 - 2 - 1 W / I 3 3 x 4 / 3 1 . f ' ( 1 ) = - 3 ( 1 ) 4 / 3 5 - W / I 2 . S' ( 7 ) = - 3 ( 7 ) 7/ 3 7 2/3 147 9. 2. Given 'f is a function such that f ( 1 ) =2 and fi ( 1 ) = 9 ch ( x ) = f ( x ) f ( x ) ." . h' ( x ) = = [ f( ) fox )] f (x ) . n ( f ( c) ) + the ( 1( x ) ) . f ( xx ) = f(x) . f'(x)+ f (x) f (ox) [ chain rule ] : . h' ( 1 ) = f ( 1) . f ' ( 1 ) + s' ( 1 ) + ( 1 ) 2 . 9 + 9. 2 36 h ' ( 1 ) = 136