f(x) = (5 sin x) / (1 + cos x)

1. Identify the rule for differentiation: Since $f(x)$ is a quotient of two functions, we use the quotient rule: if $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$. Here, $u(x) = 5 \sin x$ and $v(x) = 1 + \cos x$.

2. Differentiate the numerator and denominator:
$u'(x) = \frac{d}{dx}(5 \sin x) = 5 \cos x$
$v'(x) = \frac{d}{dx}(1 + \cos x) = -\sin x$

3. Apply the quotient rule:
$f'(x) = \frac{(5 \cos x)(1 + \cos x) - (5 \sin x)(-\sin x)}{(1 + \cos x)^2}$
$f'(x) = \frac{5 \cos x + 5 \cos^2 x + 5 \sin^2 x}{(1 + \cos x)^2}$

4. Simplify using trigonometric identities: Recall that $\sin^2 x + \cos^2 x = 1$.
$f'(x) = \frac{5 \cos x + 5(\cos^2 x + \sin^2 x)}{(1 + \cos x)^2} = \frac{5 \cos x + 5(1)}{(1 + \cos x)^2} = \frac{5(\cos x + 1)}{(1 + \cos x)^2}$
$f'(x) = \frac{5}{1 + \cos x}$

5. Calculate $f'(5)$: Substitute $x = 5$ into the simplified derivative:
$f'(5) = \frac{5}{1 + \cos 5}$