I am not sure how to solve this problem. Without seeing the function, I have no idea how to find its derivative. If there is a term I can google that will help me find out how to solve this particular derivative problem, I would be grateful to know it.
The problem states:
Let $F(x) = f(f(x))$ and $G(x) = (F(x))^2$. You also know that $f(9) = 14, f(14) = 3, f'(14) = 8, f'(9) = 3$.
Find $F'(9)$ and $G'(9)$.
How can I solve this problem?
Best Answer
$F(x)=f(f(x))$ implies $F'(x)=f'(f(x))\times f'(x)$ using the chain rule, therefore $F'(9)=f'(f(9))\times f'(9)=f'(14)\times 3=24$.
Now $G'(x)=2\times F(x)\times F'(x)$ using the chain rule again, so $G'(9)=2\times F(9)\times F'(9)=2\times f(f(9))\times 24=2\times 3\times 24=144$.