Let F ( x ) = f ( x ^2 ( f ( x ) ) where f ( 2 ) = 4 , f ( 8 ) = 5 , f ′ ( 2 ) = − 2 , f ′ ( 16 ) = 6 . Find F ′ ( 2 ) .

Question

i have no idea how to do this

Answer 1

This looks like a combination of Chain Rule and Product Rule. Use the Chain Rule to find the derivative of F(x). Since the interior of the function is a product of two functions, you'll need to use the Product Rule for your chain.

F(x) = f(x^2*f(x))

F'(x) = f'(x^2*f(x)) * (2x*f(x) + x^2*f'(x))

F'(2) = f'(2^2*f(2)) * (2(2)*f(2) + 2^2*f'(2))

= f'(4*f(2)) * (4*f(2) + 4*f'(2))

From the question, we know that:

f(2) = 4

f'(2) = -2

So,

F'(2) = f'(4*4) * (4*4 + 2*(-2))

= f'(16) * (16 - 4)

= f'(16) * 12

From the question, we know that:

f'(16) = 6

So

F'(2) = 6 * 12 = 72