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?

Use substitution: set $y=x^{\tfrac14}, \enspace k=(x+h)^{\tfrac14}-x^{\tfrac14}$. Note $k\to 0\;$ as $h\to 0$. Let's rewrite the variation rate:
$$\frac{(x+h)^{1/4}-x^{1/4}}{h}=\frac k{(y+k)^4-y^4}=\frac k{4y^3k+6y^2k^2+4yk^3}=\frac 1{4y^3+6y^2k+4yk^2},$$
which tends to
$$\frac1{4y^3}=\frac1{4x^{3/4}}$$
as $h$ (or $k$) tends to $0$.
