How $\ln{f} = a_{1}\ln{x_{1}} + a_{2}\ln{x_{2}} +...+ a_{k}\ln{x_{k}}$ is differentiated here?

Solution 1:

Consider all the $x_i$ to be functions of some parameter $t$. Since $f$ is a function of the $x_i$, this also makes $f$ a function of $t$. So the derivative with respect to $t$ is

$$\frac{d}{dt}\ln(f) = a_1\frac{d}{dt}\ln x_1 + \dots + a_k\frac{d}{dt}\ln x_k$$ $$\frac 1{f}\frac{df}{dt} = a_1\frac 1{x_1}\frac{dx_1}{dt} + \dots + a_k\frac 1{x_k}\frac{dx_k}{dt}$$ Multiplying through by $dt$, $$\frac{df}{f} = a_1\frac{dx_1}{x_1} + ... + a_k\frac{dx_k}{x_k}$$

When finite differences $\Delta f, \Delta x_i$ are very small, they behave nearly like infinitesimals. The equation you are after is making this approximation. So replace each of the $df, dx_i$ with $\Delta f, \Delta x_i$, and you get

$$\frac{\Delta f}{f} = a_1\frac{\Delta x_1}{x_1} + ... + a_k\frac{\Delta x_k}{x_k}$$