Question

f(2) = −4, g(2) = 2, f '(2) = −5 and g'(2) = 3. Find h'(2).

h(x) = g(x) / 1 + f(x)

Answer 1

You could use the quotient rule.
Let's just say \(\color{red}{g(x)=u(x)}\)
\(\color{blue}{1+f(x)=w(x)}\)

I'm just using this notation so you can see the quotient rule.
\[ h(x)=\frac{\color{red}{u}}{\color{blue}{w}}\], so by the quotient rule:
\[ h'(x)=\frac{\color{red}{u}'\color{blue}{w}-\color{red}{u}\color{blue}{w}'}{\color{blue}{w}^2}\].. substituting what you actually have:

\[\begin{align}h'(x)&=\frac{(\color{red}{g(x)})'\cdot[\color{blue}{1+f(x)}]-\color{red}{g(x)}\cdot[\color{blue}{1+f(x)}]'}{[\color{blue}{1+f(x)}]^2}\\~\\&=\frac{g'(x)[1+f(x)]-g(x)[f'(x)]}{[1+f(x)]^2}\end{align}\]
so:
\[ h'(2)=\frac{g'(2)[1+f(2)]-g(2)[f'(2)]}{[1+f(2)]^2}\]