Why is slope represented as delta y over delta x?

[vanmaiden]

Homework Statement

You always see slope represented as $\frac{\delta y}{\delta x}$. Is there any particular reason for why the change in "y" is in the numerator and the change in "x" in the denominator? Why couldn't we represent it as delta x over delta y?

Homework Equations

$\frac{\delta y}{\delta x}$

The Attempt at a Solution

I was looking at the derivative coefficient $\frac{dy}{dx}$ late last night and couldn't figure out why its delta y over delta x. Could someone fill me in on the logic behind this? why can't slope be represented as delta x over delta y?

 

[Hurkyl]

You always see slope represented as $\frac{\delta y}{\delta x}$. Is there any particular reason for why the change in "y" is in the numerator and the change in "x" in the denominator??

Er, what do you think the slope of a line is? (I'm not asking how do you think slope is computed, I'm asking what you think it actually is)

 

[vanmaiden]

Haha, I think that slope is rise over run. I guess what I'm trying to say is that why can't it be run over rise? lol.

 

[BloodyFrozen]

I think it can
i.e.

y = 1/2x
Rise 1, Run 2

x = 2y

Run 2, Rise 1

 

[HallsofIvy]

When you have y as a function of x, the derivative tells you how fast y changes relative to x.
We typically think of the dependent variable as changing because the independent variable changes and are interested in how fast the dependent variable changes relative to the independent variable.
We are seldom interested in how fast the independent variable changes relative to the dependent variable.

If, for example, y represents distance traveled, in miles, as a function of time, x, in hours, dy/dx is the "speed" in miles per hour. "dx/dy" would be hours per mile. It's a perfectly good calculation, but not one we typically use.

 

[vanmaiden]

We are seldom interested in how fast the independent variable changes relative to the dependent variable.

If, for example, y represents distance traveled, in miles, as a function of time, x, in hours, dy/dx is the "speed" in miles per hour. "dx/dy" would be hours per mile. It's a perfectly good calculation, but not one we typically use.

That is EXACTLY what I was looking for. Thank you HallsofIvy! I figured that you could calculate the rate of change of the independent variable relative to the rate of the dependent variable, but I was wondering why it wasn't used. Thanks again!

Thank you all else who helped me as well! (: