Simple Partial Differentiation Question

[Vagrant]

Homework Statement

I found this solved example in an old textbook. I don't think that the solution provided is correct. I'll be very grateful if someone could verify it.

Question:
x^xy^yz^z = c

What is $\frac{∂z}{∂x}$?

Solution Provided:
Taking logarithms on both sides:
zlog(z) = log(c) - xlog(x) - ylog(y)

Differentiating w.r.t. x
$(z.\frac{1}{z}+ log(z))\frac{∂z}{∂x} = -(x.\frac{1}{x} + 1.log(x))$

Homework Equations

The Attempt at a Solution

Isn't this solution missing a $\frac{∂y}{∂x}$ term as:
$(z.\frac{1}{z}+ log(z))\frac{∂z}{∂x} = -(x.\frac{1}{x} + 1.log(x)) - (y.\frac{1}{y} + 1.log(y))\frac{∂y}{∂x}$

Thanks.

 

[tiny-tim]

Hi Vagrant!

x^xy^yz^z = c

What is $\frac{∂z}{∂x}$?
…
Isn't this solution missing a $\frac{∂y}{∂x}$ term

ah, ∂z/∂x is defined as the derivative of z wrt x, keeping all other variables constant

[this is so even if y is also a function of x …

in that case, if you want the derivative of z wrt x to include the variation in y, you write it dz/dx not ∂z/∂x]​

see eg http://en.wikipedia.org/wiki/Partial_derivative …

" all the other variables are treated as constant when taking the partial derivative …"​

 

[Vagrant]

Ahh...ok. I get it. Thanks a lot :)