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 ________ .

Step-by-step solution

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