How to differentiate this function to find f'(7)?

Suppose that f(x) is a differentiable function such that f(7+h)=2/(h+1) and that f(7)=2. Then f'(7) = ??? The answer is -2, I'm just not sure how to reach it.

May 16, 2018

See below

Explanation:

We use the definition of the derivative

${f}^{'} \left(x\right) = {\lim}_{h \to 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

For the given function, we have

${f}^{'} \left(7\right) = {\lim}_{h \to 0} \frac{f \left(7 + h\right) - f \left(7\right)}{h}$
$q \quad = {\lim}_{h \to 0} \frac{1}{h} \left(\frac{2}{h + 1} - 2\right)$
$q \quad = {\lim}_{h \to 0} \frac{1}{h} \frac{2 - 2 h - 2}{h + 1}$
$q \quad = {\lim}_{h \to 0} - \frac{2}{h + 1}$
$q \quad = - 2$

Alternative

You could also simply recognize that the function is simply

$f \left(x\right) = \frac{2}{\left(x - 7\right) + 1} = \frac{2}{x - 6}$

so that

${f}^{'} \left(x\right) = - \frac{2}{x - 6} ^ 2 \implies {f}^{'} \left(7\right) = - 2$

I guess from the way the problem is worded, though, that a calculation using the definition of the derivative is really being sought here.