Show that the difference quotient of $1/x^n$ exists Let $n&gt;0$ be a positive integer. For all $x\not=0$, prove that $f(x) = 1/x^n$ is differentiable at $x$ with $f^\prime(x) = -n/x^{n+1}$ by showing that the limit of the difference quotient exists.


I am having trouble seeing how I can manipulate the difference quotient in order to get a limit that exists

so far I have

$$f^\prime(x)= \lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h} =\lim_{h\rightarrow 0} \frac{1/(x+h)^n - 1/x^n}{h}$$

All help is much appreciated.
Let $x\not=0$. Use the binomial theorem:

\begin{align}
\frac{\frac{1}{(x+h)^n}-\frac{1}{x^n}}{h} &amp;=-\frac{(x+h)^n - x^n}{(x+h)^n x^n h}\\
&amp;=-\frac{1}{h (x+h)^n x^n} \left(\sum_{k=0}^n {n\choose k} x^k h^{n-k} - x^n\right)\\
&amp;=-\frac{1}{h (x+h)^n x^n} \sum_{k=0}^{n-1} {n\choose k} x^k h^{n-k}\\
&amp;=-\frac{1}{(x+h)^n x^n} \sum_{k=0}^{n-1} {n\choose k} x^k h^{n-k-1}
\end{align}

Now we can safely let $h\rightarrow 0$. This makes the sum collapse to the summand with $k=n-1$ and we get

$$f^\prime(x)=-\frac{1}{x^{2n}} {n\choose n-1} x^{n-1}=-\frac{n}{x^{n+1}}$$
Collect fractions and use the binomial theorem.