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?

Hint:
Use the fact that
$$\frac{(x+h)^{1/4}-x^{1/4}}{h}=\frac{\sqrt[4]{x+h}-\sqrt[4]{x}}{h}$$
Multiply both terms in the fraction by $$(\sqrt[4]{x+h})^3+(\sqrt[4]{x+h})^2\sqrt[4]{x}+(\sqrt[4]{x+h})(\sqrt[4]{x})^2+(\sqrt[4]{x})^3$$
And simplify:
$$\lim_{h\to 0}\frac{(x+h)^{1/4}-x^{1/4}}{h}=\lim_{h\to 0}\frac{(\sqrt[4]{x+h})^4-(\sqrt[4]{x})^4}{h\left[(\sqrt[4]{x+h})^3+(\sqrt[4]{x+h})^2\sqrt[4]{x}+(\sqrt[4]{x+h})(\sqrt[4]{x})^2+(\sqrt[4]{x})^3\right]}$$
