# Weird derivative with respect to inverse temperature identity in Tong's statistical physics lecture notes

While reading David Tong's Statistical Physics lecture notes (https://www.damtp.cam.ac.uk/user/tong/statphys.html) I came across this weird identity in page 26 (at the end of the 1.3.4 free energy subsection)

$$\frac{\partial}{\partial \beta} = -k_B T^2 \frac{\partial}{\partial T}$$

whereas $$\beta$$ is as usual $$\frac{1}{k_B T}$$. I can't wrap my head around this identity, so I would appreciate any help in understanding this.

• It is just the chain rule of differentiation.
– hft
Commented Sep 16, 2023 at 23:31
• Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer.
– Community Bot
Commented Sep 16, 2023 at 23:52

...I came across this weird identity...

$$\frac{\partial}{\partial \beta} = -k_B T^2 \frac{\partial}{\partial T}$$

This is just the chain rule of differentiation.

whereas $$\beta$$ is as usual $$\frac{1}{k_B T}$$...

Yes, so that means that $$\frac{dT}{d\beta} = -k_B T^2$$

And therefore: $$\frac{df}{d\beta} = \frac{dT}{d\beta}\frac{df}{dT} = -k_B T^2 \frac{df}{dT}$$

• Thanks a lot! This is very helpful. Commented Sep 17, 2023 at 12:14