# Explain partial derivatives v3

This time I want to discuss the proof(part of it) of $$D_uf(x,y)=f_x(x,y)a + f_y(x,y)b, u = <a,b>$$
if we define a function g of the single variable h:$$g(h)=f(x_0+ha,y_0+hb)$$then, by definition of derivative, we have$$g'(0)=\lim_{{h \to 0}} \frac{g(h)-g(0)}{h} = \lim_{{h \to 0}} \frac{f(x_0+ha,y_0+hb) - f(x_0,y_0)}{h}$$
So what I understand :
If I have $$P(x_0,y_0,0) and Q(x_0+ha,y_0+hb,0) and u = <a,b>$$h represents the length of PQ(right?).
So it's clear why $$\lim_{{h \to 0}} \frac{f(x_0+ha,y_0+hb) - f(x_0,y_0)}{h}$$
But what I don't understand is when we define a function g of the single variable h, h represents the length of the vector(right?), but when we are trying to get the derivative of g(any derivative for h), what I will get is $$\frac{dg}{dh} = \lim_{{h \to 0}} \frac{g(h + h) - g(h)}{h}$$
So it makes no sense, since h is ambiguous, and it means that those h's represent different values? It seems to be an easy question, but i've been studying for whole day and probably not understanding something.
P.S. let me know If you feel like an answer worth more money

• I don't fully understand your question. And I think that you meant g(x+h)-g(x) in your last equation.

• but g is function of h, not x, that's exactly what I don't understand either

• i mean that when calculating g'(0) it equals to (lets skip limit) (g(h)-g(0))/h, so for the function g(h) in it's derivative h represents the difference of independent variable(which approaching 0), but for the same time g is function of h, so I'm little bit confused

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Kav10
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• I don't call 'h' the length of 'u' since 'u' is a unit vector, it's length is 1. I'm calling 'h' the length of projection vector, which appears when I want to get the derivative at particular point, and choose other point on some distance 'h' from it in the direction of unit vector.

• Yes. g(h) represents the value of f at a point that is h units away from the initial point (x_0, y_0) in the direction of u. So, h is indeed the distance of the "projection vector" along the direction of u. The derivative g'(0) represents the rate of change of g with respect to h when h is very close to 0. This is exactly what you need to find the directional derivative of f in the direction of u.

• So, D_uf(x, y) gives you the rate of change of f in the direction of the unit vector u at the point (x,y), and it's approximated by combining the partial derivatives of f with respect to x and y along the direction of u.

• @Kav10 yeah, I understand the logic behind it, finding derivative when distance is 0 equals to the actual rate of change. But what I still didn't get is why when I differentiate g(h) h represents the length of the projection vector, to find the derivative I have to find a limit of the distance for (g(independent var + distance) - g(independent var))/distance, so even if I receive (omit the limit) g(x + h) - g(x)/h, what does 'h' represent here, when x represents independent variable of g?

• I think there might be a slight misunderstanding about the notation and its interpretation. Let me try to clarify again.

• Kav10
+1

You define g(h) as a function that represents the value of the multivariable function f(x,y) at a point that is a distance h away from the initial point (x_0, y_0) in the direction of the unit vector u=⟨a,b⟩. This can be written as g(h) = f(x_0 + ha, y_0 + hb). Here, h represents the distance along the direction of u, and x_0 + ha and y_0 + hb represent the coordinates of the point at that distance.

• When you calculate the derivative g'(h), you are looking at how the function g(h) changes as you change the distance h. The derivative is defined as g'(h) = \lim_{{\Delta h \to 0}} \frac{g(h + \Delta h) - g(h)}{\Delta h} The notation h+Delta h means you're considering a point that is a distance h+Delta h away from the initial point.

• The expression g \frac{g(h + \Delta h) - g(h)}{\Delta h measures the rate of change of g(h) with respect to h, as you vary the distance h. In this context, h is not a variable representing the independent variable of g(h), but rather the distance along the direction of u.

• The derivative g'(h) measures how g(h) changes as you move along the direction of u by changing the distance h. Remember that the derivative g'(h) provides information about how g(h) changes as you vary h, which represents the distance along the direction defined by the unit vector u.

• LaTeX doesn't work in comments section :)

• Kav10
+1

If you're looking for the rate of change of the function f(x,y) in the direction of the unit vector u at a specific point (x_0, y_0), then you are indeed interested in the derivative D_uf(x_0, y_0), and the expression g(h) and its derivative g'(h) are tools used to help understand this concept.

• I hope this explanation clarifies things!

• that is clear for me.