How to derive the resonant frequency from the RLC circuit?
Find the input admittance


$$Y = j\omega C + \frac{1}{R+j\omega L} = j\omega C + \frac{R-j\omega L}{R^2+\omega^2 L^2}$$ 
$$Y = \frac{R}{R^2+\omega^2 L^2} + j(\omega C + \frac{-\omega L}{R^2+\omega^2 L^2}) $$

Then the Resonant Fequency is when the Imaginary component of the input admittance is zero$$Im(Y)=0$$



So $$\omega C + \frac{-\omega L}{R^2+\omega^2 L^2} = 0$$

$$\omega C = \frac{\omega L}{R^2+\omega^2 L^2} $$

$$\frac{C(R^2+\omega^2 L^2)}{L} =1 $$

$$\frac{R^2C+\omega^2 CL^2}{L} =1 $$

$$\frac{R^2C}{L}+\omega^2 LC =1 $$

$$\omega^2 LC = 1 - \frac{R^2C}{L} $$

I bet you can take it form here