Where Does This Equation Originate?

[pattisahusiwa]

I have an equation like this,

$\frac{dZ}{zD\beta} = \frac{d}{d\beta}\ln Z$,

is it from $\frac{d}{d\beta}\frac{dZ}{Z}$ or from...?

How we can prove this relation?

 

[Office_Shredder]

Is your equation supposed to be
$$\frac{1}{Z} \frac{ dZ}{d\beta} = \frac{d}{d\beta} \ln(Z)$$
If so, this is just the chain rule

 

[pattisahusiwa]

Thank you for quick replay.

Yes, your relation is correct too. If this is a chain rule, so can i write them like one in the first thread?

 

[pattisahusiwa]

Hi all, I just want to know that my relation is correct or not?

$$\frac{1}{Z}\frac{dZ}{d\beta} = \frac{d}{d\beta}\int\frac{dZ}{Z} = \frac{d}{d\beta}\left(\ln Z\right)$$

 

[blue_raver22]

This equation is known as the logarithmic differentiation formula. It is derived from the chain rule in calculus. The left side of the equation represents the derivative of Z with respect to both z and \beta, while the right side represents the derivative of the natural logarithm of Z with respect to \beta. This relationship can be proven by taking the derivative of both sides of the equation and using the properties of logarithms. It is a useful tool in calculating derivatives of functions that involve complex expressions or multiple variables.