If f(1)=1, f^ (1)=3 , then the derivative of f(f(f(x)))+(f(x))^2 at x=1 is

Step-by-step solution

Let g(x) = f(f(f(x))) + [f(x)]^2 and evaluate g'(1) given f(1) = 1 and f'(1) = 3 . Differentiate g(x) using the chain rule: g'(x) = f'(f(f(x))) f'(f(x)) f'(x) + 2 f(x) f'(x) At x = 1 , applying f(1) = 1 gives: g'(1) = f'(f(1)) f'(1) f'(1) + 2 f(1) f'(1) = f'(1)^3 + 2 f'(1) Substitute f'(1) = 3 : g'(1) = 3^3 + 2 3 = 27 + 6 = 33 Final answer: 33