How do you differentiate function like x^(1/4) or x^(3/5) ect?
Given $\displaystyle f(x)= x^{\alpha}$, where $\displaystyle \alpha$ is an arbitrary real or even complex constant, its derivative by definition is...
$\displaystyle \displaystyle f^{'}(x) = \lim_{\delta \rightarrow 0} \frac{(x+\delta)^{\alpha} - x^{\alpha}}{\delta}$ (1)
Now is...
$\displaystyle \displaystyle {(x+\delta)^{\alpha} = x^{\alpha} + \alpha\ \delta\ x^{\alpha-1} + \frac{\alpha\ (\alpha-1)}{2}\ \delta^{2}\ x^{\alpha-2} + ...$ (2)
... so that, combining (1) and (2), we find that is...
$\displaystyle \displaystyle f^{'}(x)= \alpha\ x^{\alpha-1} $ (3)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$