Continuity and Differentiability · Question 4

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(x2+2x+4), and g(x)= 41+e-2x. Define the composite function f g-1 by (f g-1)(x)=f (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 .

Answer

0.2 to 0.3

Solution

(f g-1)'(2)= f'(g-1(2))g'(g-1(2)).

g(x)=2 e-2x=1 g-1(2)=0.

f'(x)= 2x+2x2+2x+4 f'(0)= 12; g'(x)= 8e-2x(1+e-2x)2 g'(0)=2.

Required value = 1/22= 14=0.25

Answer: 0.25