Let R denote the set of all real numbers. Let f: R R and g: R (0,4) be functions defined by f(x)= _e (x^2+2 x+4 ), and g(x)= 4 1+e^ -2 x Define the composite function f g^ -1 by (f g^ -1 )(x)= (g^ -1 (x) ), where g^ -1 is the inverse of the function g. Then the value of the derivative of the composite function f g^ -1 at x=2 is ________ .
A. A
aligned & Let h ( x )= f ( g ^ -1 ( x ) ) and g (0)=2 \\ & aligned h ^ ( x ) & = f ^ ( g ^ -1 ( x ) ) ( g ^ -1 ( x ) )^ \\ h ^ (2) & = f ^ ( g ^ -1 (2) ( g ^ -1 )^ (2) . \\ & = f ^ (0) ( g ^ -1 )^ (2) aligned aligned Now aligned & f(x)= _0 (x^2+2 x+4 ) \\ & f^ (x)= 2 x+2 x^2+2 x+4 \\ & f^ (0)= 1 2 \\ & g(x)= 4 1+e^ -2 x , g(0)=2 \\ & g^ -1 (g(x))=x aligned array l|l ( ( g ^ -1 )^ ( g ( x )) ) g ^ ( x )=1 & ~g ( x )= 4 1+ e ^ -2 x \\ ( g ^ -1 )^ (2)= 1 ~g ^ (0) & g ^ ( x )= 8 e ^ -2 x (1+ e ^ -2 x )^2 \\= 1 2 & ~g ^ (0)= 8 4 =2 \ h (2)= 1 4 =0.25 & array
Related: Mathematics — Differentiation · All PYQ Banks