Let f (x)= e ^x, ~g (x)= ⁻¹ x and h (x)= f ( g (x)) , then ( h ^ (x) h (x) )^2 is equal to

Step-by-step solution

aligned h (x) & = f ( ~g (x)) & = f ( ⁻¹ x ) h (x) & = e ^ ⁻¹ x aligned Differentiating w.r.t. x , we get aligned h ^ (x) & = e ^ ⁻¹ x ~d ~d x ( ⁻¹ x ) & = e ^ ⁻¹ x 1 1-x^2 aligned Now, h ^ (x) h (x) = e ^ ⁻¹ x 1 1-x^2 e ^ ⁻¹ x = 1 1-x^2 ( h^ (x) h(x) )^2= 1 1-x^2