How do you find the derivative of xth root of x?

How would you find the derivative of $\sqrt[x]{x}$? The derivative rule of ${x}^{a}=a{x}^{a-1}$ gives me the new formula of $\frac{1}{x}{x}^{\frac{1}{x}-1}$ (because $\sqrt[x]{x}={x}^{\frac{1}{x}}$). However, this equation isn't the derivative of $\sqrt[x]{x}$, and I can't figure out how else you would determine that.