Radical Differentiation Rule
I'm solving some differentiation problems and a pattern has come to my attention. I'd like to know if the rule I've come up with is true and if it is provable. It goes as follows: Given a function $f$ with $f(x)=\sqrt[n]{x}$ then $f'(x)=\frac{1}{n\sqrt[n]{x^{n-1}}}$. Thanks in advance.
Solution 1:
$f(x)=x^{1/n}$ so $f'(x)=\frac{1}{n}x^{1/n-1} =\frac{1}{nx^{1-1/n}}$, equivalent to your expression. How'd you get your expression? It's very useful to derive it from the definition of the derivative. There might be some other ways, e.g. by induction.