Simple Partial Differentiation Question

In summary, the conversation is about a provided solution to a problem involving xxyyzz = c and the question of what the partial derivative of z with respect to x is. The solution provided is questioned for potentially missing a term involving the partial derivative of y with respect to x. It is clarified that the solution is correct as it only considers the partial derivative of z with respect to x while keeping all other variables constant.
  • #1
Vagrant
195
1

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:
xxyyzz = c

What is [itex]\frac{∂z}{∂x}[/itex]?

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

Differentiating w.r.t. x
[itex](z.\frac{1}{z}+ log(z))\frac{∂z}{∂x} = -(x.\frac{1}{x} + 1.log(x))[/itex]

Homework Equations




The Attempt at a Solution



Isn't this solution missing a [itex]\frac{∂y}{∂x}[/itex] term as:
[itex](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}[/itex]

Thanks.
 
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  • #2
Hi Vagrant! :smile:
Vagrant said:
xxyyzz = c

What is [itex]\frac{∂z}{∂x}[/itex]?

Isn't this solution missing a [itex]\frac{∂y}{∂x}[/itex] term


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

[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 …"​
 
  • #3
Ahh...ok. I get it. Thanks a lot :)
 

1. What is simple partial differentiation?

Simple partial differentiation is a mathematical concept used to find the rate of change of a function with respect to a specific variable, while holding all other variables constant. It is often used in multivariable calculus to analyze how a function changes in response to small changes in one of its variables.

2. What is the difference between partial differentiation and total differentiation?

The main difference between partial differentiation and total differentiation is that partial differentiation only considers the rate of change with respect to one variable, while total differentiation takes into account the rates of change with respect to all variables in a function. Total differentiation is also known as the total derivative.

3. How is simple partial differentiation performed?

To perform simple partial differentiation, you first identify the variable you want to differentiate with respect to. Then, you differentiate the function with respect to that variable, treating all other variables as constants. This results in a new function, known as the partial derivative, which represents the rate of change of the original function with respect to the chosen variable.

4. What are some real-world applications of simple partial differentiation?

Simple partial differentiation has many applications in fields such as physics, economics, and engineering. It is used to analyze how a system responds to changes in one of its variables, such as how the temperature of a chemical reaction affects its rate, or how changes in interest rates impact the value of a stock portfolio.

5. Can simple partial differentiation be applied to non-linear functions?

Yes, simple partial differentiation can be applied to non-linear functions. The process is the same as for linear functions, but the resulting partial derivatives may be more complex. In cases where the partial derivatives cannot be easily calculated, numerical methods may be used to approximate them.

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