Let f(x) and g(x) be continuous and differentiable functions such that f(1) = f'(1) = 2, f'(3) = 4, g(2) = 3, and g'(1) = 5. Define F(x) = f(g(xf(x))

Question

Let f(x) and g(x) be continuous and differentiable functions such that f(1) = f'(1) = 2, f'(3) = 4, g(2) = 3, and g'(1) = 5. Define F(x) = f(g(xf(x))

Please help none of my friends can get this question.

Answer 1

The answer is 48. Here's why- when you have a composite function, you must apply the chain rule. The rule follows three steps:

1) The derivative of the outside,

2) Evaluated at the inside,

3) Times the derivative of the inside.

If F(x) = f(g(xf(x))), then F'(x) = f'(g(xf(x))*g'(x*f(x))*(f(x)+ xf'(x)))

plugging in each of the values you have above results in 48. This is a complicated problem. I would be happy to explain it in much greater detail if interested.