First, we need to compute the general derivative of F using the chain rule and the product rule.
Essentially, the chain rule says that the derivative of a function like f(g(x)) is f'(g(x))g'(x). In words, "Take the derivative of the 'outside' function, leave the 'inside' function the same. Multiplied by the derivative of the 'inside' function."
The product rule says that the derivative of the product of two functions f(x)*g(x) is f'(x)*g(x) + f(x)*g'(x). In words, "The derivative of the 'first' function times the 'second' function plus the 'first' function times the derivative of the 'second' function."
Now, we see F(x) = f(xf(x^2)). Using the chain rule,
F'(x) = f'(xf(x^2))*(xf(x^2))'
Next, we use the product rule to compute (xf(x^2))':
(xf(x^2))' = x*(f'(x^2)*2x) + f(x^2)*1 = (2x^2)*f'(x^2)
Putting this all together,
F'(x) = f'(xf(x^2))*(2x^2)*f'(x^2)
Now, we plug in x=2 to compute F'(2):
F'(2) = f'(2*f(2^2))*(2*(2^2))*f'(2^2)
= f'(2f(4))*(2*4)*f'(4)
= f'(2*4)*8*1
= f'(8)*8
= 2*8
= 16
