Clarification of the independent variable for a partial derivative

[Dorslek]

For some non-linear 3D function, let's say I want to take the partial derivative for x where y is constant. Each point for Z will be different of course since it's non-linear.

So let's say I plug in an X of 3 where Y is constant for some function and I get a slope of 5 as my answer. Is it correct to interpret it as:

"If I move a very small amount away from 3 on the x-axis, the Z-axis will be 5 times as large as that very small amount and thus you have your tangent line"

So the concept I'm trying to make sure I have down 100 percent is that I always interpret the change in X as a very small amount and the change in Z as the rate of change relative to this very small change correct?

I couldn't in my example use one unit away or twenty units away from 3 since the answer can be drastically different from my rate of change at 3 based on the very small amount away from 3.

This clarification was brought about not only to make sure I'm thinking about this right but also in the fact that when you use ddx/ddy in shader functions, their very small change is always one pixel to the right/up/down/left away.

 

[HallsofIvy]

I'm not particularly comfortable with talking about "infinitesmals" since the logical basis needed is pretty complicated (and 1 "pixel" is not normally what you mean by "infinitesmal"). However, it is true that the line [itex]x= X+ t[/itex], [itex]y= Y[/itex], [itex]z= f(X, Y)+ f_x t[/itex] and the line [itex]x= X[/itex], [itex]y= Y+ t[/itex], [itex]z= f(X, Y)+ f_yt[/itex] are tangent to the surface z= f(x,y) at (X, Y, F(X, Y))

 

[Dorslek]

I'm not particularly comfortable with talking about "infinitesmals" since the logical basis needed is pretty complicated (and 1 "pixel" is not normally what you mean by "infinitesmal"). However, it is true that the line [itex]x= X+ t[/itex], [itex]y= Y[/itex], [itex]z= f(X, Y)+ f_x t[/itex] and the line [itex]x= X[/itex], [itex]y= Y+ t[/itex], [itex]z= f(X, Y)+ f_yt[/itex] are tangent to the surface z= f(x,y) at (X, Y, F(X, Y))

Edited that out. I can then assume then if no units are mentioned, such as a pixel, that a partial derivative for z in x then is always going to be a very small change in X just like when doing a regular derivative?