Integration by parts, don't quite know how to arrive at the given answer

[DiffusConfuse]

I am assuming that the solution was arrived at through integration by parts, however I am not able to completely work through it.

First given: c_B= X_B/V_m

the next step shows the solution to dc_B given as:

dc_B=(1-dlnV_m/dlnx_B)(dx_B/V_m)

 

[HallsofIvy]

I have no clue what you mean here. "d" usually indicates a derivative, not an integral. Could please specify what is the entire problem here.

 

[DiffusConfuse]

That is the entire problem, it was just written in a paper as such. I am aware that d usually means derivative. It shows dc is used in a diffusion problem and here the aim is to use concentration data that is given in fraction form i.e. mole fraction over volume.

 

[phyzguy]

It's just applying the chain rule and collecting terms. First, notice that:
$$d(ln(x)) = \frac{dx}{x}$$
Then, apply the chain rule to the original differentiation:
$$d(c_B) = d(\frac{x_B}{v_m}) = \frac{dx_B}{v_m}-\frac{x_B dv_m}{v_m^2} = \frac{dx_B}{v_m}(1-\frac{x_B}{v_m}\frac{dv_m}{dx_B}) = \frac{dx_B}{v_m}(1-\frac{d(ln(v_m))}{d(ln(x_B))})$$

 

[DiffusConfuse]

thank you, did not remember that identity