I'm currently trying to prove the power rule using the epsilon-delta definition of a derivative. I've already done it for the basic limit definition, but I thought it might be a helpful exercise to test my understanding by doing it this way. However, I'm struggling and would really appreciate any help/hints.
My work so far:
The epsilon-delta definition says a function is differentiable if, for every $\epsilon > 0$, there exists a $\delta > 0$ such that $|x-x_0| < \delta$ implies $\frac{f(x)-f(x_0)}{x-x_0} – f'(x_0) < \epsilon$. Thus, I need to find a delta, as function of epsilon, and possibly $x_0$, such that whenever the first inequality is true the second one is as well.
So, fix $x_o \in R$, and let $|x-x_0| < \delta$, then $|\frac{x^n – x_{0}^{n}}{x-x_0} – nx_{0}^{n-1}| = |\frac{x^n – nxx_{0}^{n-1} + (n-1)x_{0}^{n}}{x-x_0}|$.
Upper-bounding this with $x < \delta + x_0$ and simplifying, we get $|\frac{(\delta + x_0)^n – n\delta x_{0}^{n-1}-x_{0}^{n}}{\delta}| = |\frac{(\delta + x_0)^n}{\delta} -nx_{0}^{n-1} – x_{0}^{n}| < \epsilon$.
Is that right so far, and if so, any advice on how to push through this last step and solve for $\delta$? I'm stumped, to be honest.
-Thank you in advance for your help!
Best Answer
Using
$$(x^n - x_0^n) = (x - x_0)\sum_{k=0}^{n-1} x^k x_0^{n-1-k}$$
One can write
$$ \frac{x^n - x_0^n}{x - x_0} - n x_0^{n-1} = \sum_{k=0}^{n-1} (x^k x_0^{n-1-k} - x_0^{n-1}) $$ Define $M = |x_0| + 1$ and suppose that $|x-x_0|\le 1$, then $|x|\le M$ and $|x_0|\le M$ and $$ |x^k-x_0^k| \le |x - x_0|\sum_{p=0}^{k-1} |x|^p |x_0|^{k-1-p}\le |x - x_0|\sum_{p=0}^{k-1} M^{k-1}\le |x-x_0|k M^{k-1} $$
We get $$ \left|\frac{x^n - x_0^n}{x - x_0} - n x_0^{n-1}\right| \le \sum_{k=1}^{n-1} |x-x_0|k M^{k-1} |x_0|^{n-1-k}\le |x - x_0|M^{n-2}\sum_{k=1}^{n-1}k \le |x - x_0|M^{n-2}\frac{n (n-1)}{2} $$
For $n>1$ we can take $$\delta = \min\left(1, \frac{2\varepsilon}{n(n-1)(|x_0|+1)^{n-2}}\right)$$ For $n=1$ there is no condition on $\delta$ because $f(x) = f(x_0) + (x-x_0)$.