If f(x)= ⁻¹ ( 2^ x+1 1+4^x ) then f^ (0) is equal to

Step-by-step solution

Given f(x) = ⁻¹ ( 2^ x+1 1+4^x ) . Let 2^x = . Then 4^x = (2^x)^2 = ^2 . The expression inside the inverse sine becomes 2 2^x 1 + (2^x)^2 = 2 1 + ^2 = (2 ) . Thus, f(x) = ⁻¹( (2 )) = 2 = 2 ⁻¹(2^x) . Differentiating with respect to x using the chain rule: f'(x) = 2 1 1 + (2^x)^2 d dx (2^x) = 2 1 + 4^x 2^x 2 . Evaluating at x = 0 : f'(0) = 2 2^0 2 1 + 4^0 = 2 1 2 1 + 1 = 2 2 2 = 2 . Answer: 2