Chemical Forums
Specialty Chemistry Forums => Other Sciences Question Forum => Topic started by: zeshkani on August 30, 2007, 01:49:21 PM

can somebody explain on how to do a total differentiation on this, i have the formula but iam just lost
z=x^2+2y^22xy+2x4y8

You can look at this term by term to make things simpler. I'll do the first couple of terms to start things off:
x^{2}
The differential of this term is fairly easy since it involves only one variable. I like to think of a differential this way. You know the derivative (dz/dx) of z = x^{2}:
(dz/dx) = 2x
Well, if you treat the derivative like a fraction, and multiply through by dx, you get:
dz = 2x dx
which is the differential of x^{2}.
2xy
This term is a more difficult term. Here you have to make use of the product rule:
d(uv) = u dv + v du
So in this case,
d(2xy) = 2 d(xy) = 2(xdy + ydx) = 2x dy  2y dx
You should be able to do the rest now.

would this be the answer:
dz=2x2y+2dx + (4y2x4 dy)

Do you mean:
dz = (2x2y+2) dx + (4y2x4) dy

yeah thats what i mean i just forgot to put () around the dx part

if you don't want to think like Yggdrasil you could form the partial derivatives:
dz/dx = 2x  2y + 2
dz/dy = 4y  2x  4
then the differential is defined as:
dz = (2x  2y + 2)dx + (4y  2x  4)dy

thx for all the help, i finally got it ;D

FeLiXe's method is probably the easier way to think of things. Conceptually, though it is important to know the product rule for differentials (i.e. d(uv) = u dv + v du).