Derivatives of Functions

Question

Suppose $ F(x)=f(g(x)) $

• $g(1)=3$
• $g'(1)=4$
• $f'(1)=6$
• $f'(3)=5$

What is $F'(1)$ ?

Answer 1

Hint: By the Chain Rule, $F'(x)=g'(x)f'(g(x))$. Now use the information provided to evaluate the various bits when $x=1$.

Answer 2

$F(x)=f(g(x))$, so $F'(x)=f'(g(x))g'(x)$ and for $x=1$ it is $F'(1)=f'(g(1))g'(1)=f'(3)g'(1)=5\cdot4=20$

Answer 3

$$F'(x) = f'(g(x))\cdot g'(x)$$ $$F'(1) = f'(g(1))\cdot g'(1) = f'(3)\cdot 4 = 5 \cdot 4 = 20$$