By definition of exponentiation, $\displaystyle 1^{0.5}=1^{1/2}=\sqrt{1}$, which is 1.

What exactly is f(x)? Do you mean that the derivativeoff(x) is $\displaystyle \frac{2}{3}x^{2/3}$, or do you want to say that $\displaystyle f(x)=\frac{2}{3}x^{2/3}$ is a derivative of something else? My guess is that $\displaystyle f(x)=\sqrt{x}=x^{1/2}$ and itsantiderivative $\displaystyle F(x)=\frac{2}{3}x^{3/2}$ (plus a constant). This is so because $\displaystyle \left(\frac{2}{3}x^{3/2}\right)'=x^{1/2}$ according to the power rule.

In plain text, you can write x^(2/3) to mean $\displaystyle x^{2/3}$. Note the parentheses; x^2/3 means $\displaystyle \frac{x^2}{3}$.