Derivatives of composite functions

Question

How would I solve a problem that is asking me to find the derivative of $F$ when $$F(x)=f\left(\frac{x+2}{x+4}\right)$$ and $f$ is differentiable.

Not asking for the answer here obviously, just the steps needed to get off the ground.

Answer 1

$$F(x)=f(\frac{x+2}{x+4})$$

$$F'(x)=f'(\frac{x+2}{x+4})(\frac {x+2}{x+4})'=f'(\frac{x+2}{x+4})(\frac {(x+4)-(x+2)}{(x+4)^2})=(\frac {2}{(x+4)^2})f'(\frac{x+2}{x+4})$$

Answer 2

You use the chain rule.

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

then $F'(x) = g'(x)f'(g(x))$

Answer 3

$$ F'(x)=f'\left(\frac{x+2}{x+4}\right)\cdot\left(\frac{x+2}{x+4}\right)'=f'\left(\frac{x+2}{x+4}\right)\cdot\frac{2}{(x+4)^2} $$