Obtaining a derivative using limit definition We have the following limit $$ \lim_{h \to 0} \dfrac{(x+h)^{1/4}-x^{1/4}}{h}$$
I want to find this limit (which I figure is just the derivative of $x^{1/4}$) using only elementary methods (algebra, mostly).
So you can rewrite the function as $\dfrac{4}{h}(\sqrt{x+h} - \sqrt{x})$, but you still have that $h$ in the denominator which makes it impossible to take the limit. I can't seem to rewrite this in a way that I get the expected answer, can anyone give a hand?

Using Binomial Expansion,
 $\frac{(x+h)^{1/4}-x^{1/4}}{h} = \frac{x^{1/4}+(1/4)x^{(1/4-1)}+O(h^2)-x^{1/4}}{h}=\frac{(1/4)x^{(1/4-1)}+O(h^2)}{h} = (1/4)x^{(1/4-1)}+O(h)$
$lim_{h\rightarrow0} (1/4)x^{(1/4-1)}+lim_{h\rightarrow0}O(h) = (1/4)x^{(1/4-1)} = (1/4)x^{-5/4} $
