Finding the max frequency of a driven oscillator

[SuchBants]

So I've derived the equation for the amplitude of a driven oscillator as:

$\huge A=\frac{F}{m\sqrt{(\omega_{0}^{2}-\omega_{d}^{2})^{2}+4\gamma^{2}\omega_{d}^{2}}}$

Which is what my lecturer has written. Then taking the derivative and setting it to 0 to get the turning point. He makes this leap:

https://imgur.com/a/gE7Y0Di

How does he do that? I can't do it.

Also an auxiliary question. I was watching Walter Lewin on this here .
And he uses

$\huge {\gamma}=\frac{b}{m}$

Whereas I've been taught:

$\huge {\gamma}=\frac{b}{2m}$

Where ϒ is the damping coefficient. Which is correct?

 

[Charles Link]

His definition is ok. Both are used for $\gamma$. $\\$ If you take the derivative $\frac{dA}{d \omega_D}$ and set it equal to zero, you get the peak of the $A$ vs. $\omega_D$ graph,(the resonant frequency), which peaks just slightly off from $\omega_o=\sqrt{k/m}$. The derivative is quite a simple operation with the chain rule. $\\$ Note: He starts with $F(t)=Fe^{i \omega_D t}$, and computes the amplitude $A=|x|$.