Binomial expansion derivative limit definition Can someone help me with this? I am supposed to use a binomial expansion to calculate $\sqrt x$ directly from the limit definition of a derivative.
 A: The definition of the derivative
$$f'(x) = \lim_{h\to0} \frac {f(x+h)-f(x)}h$$
Here $ f(x) = \sqrt{x} $
$$f'(x) = \lim_{h\to0} \frac {\sqrt{x+h}-\sqrt{x}}h$$
Rationalize:
$$f'(x) = \lim_{h\to0} \frac {\sqrt{x+h}-\sqrt{x}}h \cdot \frac {\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}$$
And the rest should be straightforward...
A: I remain uncertain what is meant by "use a binomial expansion".  There is a binomial expansion of $\sqrt x = \sqrt{1+(x-1)}$ in powers of $x-1$, and that is an infinite series, but you wouldn't be using the definition of "derivative" directly if you did that.  Perhaps the expansion of the product of two sums in the answer posted by "ImATurtle" can be considered a "binomial expansion" since it's an expansion of the product of two binomials.
Here's a similar method:
\begin{align}
\frac d{dx} \sqrt x & = \lim_{\Delta x\to 0} \frac{\Delta \sqrt x}{\Delta x} = \lim_{w\to x} \frac{\sqrt w - \sqrt x}{w - x} = \lim_{w\to x} \frac{\sqrt w - \sqrt x}{(\sqrt w - \sqrt x)(\sqrt w + \sqrt x)} \\[10pt]
& = \lim_{w\to x} \frac 1 {\sqrt w + \sqrt x} = \frac 1 {\sqrt x + \sqrt x}.
\end{align}
