# Differentiation of coordinate wrt another coordinate

When I take the differential of y wrt t (being t a parameter (time)) I get the velocity of the y-coordinate, if take the second differential of y wrt t, thus I get the aceleration of the y-coordinate... ok! But what means to differentiate the y-coordinate wrt x-coordinate, or wrt y, or then differentiate twice wrt to y, or x, or xy. ?

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lurflurf
Homework Helper
Do you mean the derivative or the differential?
A common situation would be if the movement is constrained so that
f(x,y,t)=0
by the Implicit function theorem we can think of x and y as each being functions of time, or as being functions of each other.

Do you mean the derivative or the differential?
differential in the inverse sense of integral, ie, "the differential of f wrt to x results the derivative of f"...

lurflurf has answered the question nicely.

I would add the following link, which is the most beautiful explanation of partial derivatives which I've encountered.
http://www.av8n.com/physics/partial-derivative.htm
His answer did not clear anything to me. You text-link is very big, gives to summarize?

It is good to picture the situation: What you have is the plane ##\mathbb R ^2## and points ##(x,y)## on that plane. Then you put a curve ##\gamma## into that plane via
##
\gamma(t)= (x(t), y(t)) \, .
##
Taking the derivative with respect to time gives you the velocity $$\dot \gamma (t)$$. Getting rid of the parameter by solving y for x gives you the (image of the) curve without telling you how fast you go along that curve. Then differentiating ##y## with respect to ##x## tells you the slope of that curve at point ##(x, y(x))##. For simplicity I assumed there is only on ##y## value corresponding to a given ##x##-value.

His answer did not clear anything to me. You text-link is very big, gives to summarize?
From the top of the link:

Executive summary: Partial derivatives have many important uses in math and science. We shall see that a partial derivative is not much more or less than a particular sort of directional derivative. The only trick is to have a reliable way of specifying directions ... so most of this note is concerned with formalizing the idea of direction. This results in a nice graphical representation of what “partial derivative” means. Perhaps even more importantly, the diagrams go hand-in-hand with a nice-looking exact formula in terms of wedge products.

My summary: If you really want to understand partial derivatives, this link is a good reference.