# Differentiation of coordinate wrt another coordinate

#### Jhenrique

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.

#### chogg

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

#### Jhenrique

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?

#### Geometry_dude

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.

#### chogg

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.

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