How Does the Total Derivative Sum Up Changes in Multiple Directions?

[Agent 47]

##dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy##

I'm confused as to how the total derivative represents the total change in a function.

My own interpretation, which I know is incorrect, is that ##\frac{\partial z}{\partial x} dx## represents change in the x direction and ##\frac{\partial z}{\partial y} dy## change in the y direction, and to get the total change you would need to square both and take the square root much like finding the magnitude of a vector.

Obviously this is not the case because, as the equation shows, you simply need to add the two terms together.

So my overarching question is: Could someone please intuitively explain what the total differential represents and why it's a simple sum?

 

[WWGD]

The total derivative is the best linear approximation to the (local) change in values of a differentiable function.

The case you use in your post is that of a tangent plane. Given a 2-dimensional object Y:=f(X) embedded in R^n, if f is differentiable at a point y in f(X), this is equivalent to saying that , near y=f(x), the change ##f(x)-f(x_0) ## can be
(locally, unless f is itself linear) approximated to any degree of accuracy, in a delta-epsilon sense, by a linear function L(x_o, \epsilon); for Y 2-dimensional , L is the plane tangent to Y at x=x_o.

 

[Agent 47]

Then my followup question would be how does the addition of the two tangent lines actually equate to the correct tangent plane. Is there a proof for tangent plane approximation that doesn't use the chain rule?

 

[WWGD]

I don't understand what you mean by the addition of tangent lines. How is this addition defined?

 

[Agent 47]

Sorry for the vagueness. Here is what I'm referring to:

The way it's presented in my textbook makes it seem like the addition of two tangent lines lying on the same plane. What I'm wondering is how adding two lines lying in the same plane actually gives you that plane.

 

[WWGD]

But notice you are not really adding lines, because lines have a fixed slope; neither ## f_y(x_0,y_0)## nor ##f_x(x_0,y_0) ## are constant; they depend on the choice of ##(x_0,y_0)##.

 

[davidmoore63@y]

I would answer your original post by saying that z is a scalar function of (x,y). The change in z has no direction, so your vector anologue is inappropriate. The equation says that if you know how the scalar z changes when you increase x by dx, and you separately know what happens to z when you increase y by dy, then if you want to know what happens if you increase both x and y, you add the two effects (at least locally when dx and dy are very small).

 

[WWGD]

##dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy##

I'm confused as to how the total derivative represents the total change in a function.
<Snip>

So my overarching question is: Could someone please intuitively explain what the total differential represents and why it's a simple sum?

The (total) derivative does not represent the total change of a function. The differential is an approximation to the total change. The best intuition I can think of is that the change of a differentiable function looks locally like the change of a linear object ( line, plane, higher-dimensional equivalent ).