Expert: Steve Holleran - 3/6/2007

Question
For my chemical engineering homework assignment, we were given an equation we need to differentiate in order to prove when a certain product recovery is maximized. I am familiar with how to find the critical points and points of inflection etc. but the equation we were given for this homework assignment was rather difficult and I'm not sure how to go about differentiating.

Here is the whole problem: Please don't let the chemical engineering context bother you!

#2: The previous problem is based on the assumption that the wash water is divided equally among the various washes.  It is legitimate to ask if this is the best way to distribute the wash water.  Let us assume the total amount of solvent is fixed and the fraction of solvent fed to a particular extraction is Fi, show that X1/Xn+1 = (Product) i [1/(1+Ei)], where Ei = KFiL/H. and Xn/Xn+1 reaches minimum if Fi = Fn = 1/N.  Therefore, you just prove that the recovery will be maximized when the wash water is divided uniformly among the various washes.  

I used lowercase to represent subscripts. i hope this isnt terribly confusing. the "(Product)" is one of those big uppercase Pi symbols that i think represents a product. My main problem is how to differentiate a (Product). If you could show me how to do this i would really appreciate it! Thanks,
Chelsea

Answer
Hi Chelsea,

Well, I think I know what you're asking, but I have to admit, I'm a little lost with all the variables, especially what the X1 and Xn+1 represent.  Also, are K and L and H constants?

In any case, what I CAN tell you is that to differentiate a product, you generally multiply each function in the product by the derivative of the other function.  In notation, if there are two functions, f and g, it goes:

(fg)' = fg' + gf'

If there are three, you have:

(fgh)' = fgh' + fg'h + f'gh

Likewise, for four:
(fghk)' = fghk' + fgh'k + fg'hk + f'ghk.

The uppercase Pi symbol you mention does indeed represent the product, just like a sigma symbol represents a sum.

I'm really sorry that I can't be of any help beyond this.
Maybe if I had a better understanding of the variables I could do better.

Good luck to you!
Steve Holleran