What is a Partial Differential, "∂" ?

Question

Can someone please explain to me what EXACTLY the partial differential means?? I haven't done calculus since high school and this ∂ symbol is confusing the hell out of me in my chemistry major. I can't seem to imagine what is actually happening when this ∂ symbol appears.
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Thanks!

Thanks, Tom and pablo, I get it a bit better now, but I was maybe looking for a more graphical explanation? Thanks a lot =)

Pablo, so the ∂g / ∂x is the same as dg / dx, when y is kept constant??

Ok, tom, but suppose i draw a graph of y against x. dy / dx is the slope of the graph (or some other thing if it is a curve ). Now if z is also a variable, and I keep z constant, then ∂y / ∂x will be equal to what dy / dx was in the other graph, considering the relationship between y and x is changed?

Ok, tom, but suppose i draw a graph of y against x. dy / dx is the slope of the graph (or some other thing if it is a curve ). Now if z is also a variable, and I keep z constant, then ∂y / ∂x will be equal to what dy / dx was in the other graph, considering the relationship between y and x is unchanged?

oopsy, double post. the 1st post is wrong, read the 2nd one, he he he =)

Answer 1

Check out the following links. They are pretty self-explanatory.

Look at formula (2) in the Exact differential link to show you the relationship between total and partial derivatives.

Of course the partial derivative link will give the definition of a partial derivative.

Migrant: Yes, that is quite correct.

Answer 2

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Answer 3

When you have a function f(x), you can always take the derivative with respect to the independent variable x. That is called a total derivative, and is written as df/dx.

When you have a funciton g(x,y), you now have to specify with respect which independent variable you are differentiating. So for our function g, there are two possible derivatives you can take, ∂g/∂x (treating y as a constant) and ∂g/∂y (treating x as a constant). These are called partial derivatives, and are given the symbol ∂.

Answer 4

It means take the derivative with respect to one of the variables and treat the other variable as if it were a constant. Example:
f(x,y) = x² + xy + y² then
∂f(x,y)/∂x = 2x + y + y² and
∂f(x,y)/∂y = x² + x + 2y

Basically, it's the derivative taken along one of the variables axis. If it's expressed as something such as
z = f(x,y) then you'd have ∂z/∂y = ∂f(x,y)/∂y and so on.

Very convenient things, but easy to misuse.

Hope that helps.


Doug