I'm trying to decipher a step undertaken to demonstrate the formula that:

[tex]A = - \frac{ln(Q_{NVT})}{ \beta } = - k_{B}Tln(Q_{NVT})[/tex]

One of the steps undertaken is:

[tex]A = E - TS[/tex]

[tex]\frac{dA}{dT}_{N,V} = -S[/tex]

Since my professor's notes defined the Boltzmann factor as [tex] \beta = \frac{1}{k_{B}T} [/tex]

He then wrote that:

[tex]\frac{dA}{dT}_{N,V} = \frac{dA}{d\beta}_{N,V} \frac{ \beta }{T} = -S[/tex]

I just don't understand how he obtained that last one... I don't think he just substituted T with 1/βk_{B}. As someone who lacks basic knowledge of mathematics, I have no idea how we arrived at that last part....