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### Course: Multivariable calculus>Unit 2

Lesson 1: Partial derivatives

# Formal definition of partial derivatives

Partial derivatives are formally defined using a limit, much like ordinary derivatives.  Created by Grant Sanderson.

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• , Is the f(a,b) also being divided by h or is it outside of the fraction?
• The f(a,b) is divided by h. This formula is based on what was covered in to , and is very similar to the df/dx formula he derived. If you remember, the concept of this general formula is centered around dy/dx. dx, of course is h, while dy is equivalent to the numerator in the formula.
• why we do partial differentiation?
• To see how how change in a single variable of a multivariable equation affects said multivariable equation's slope or rate of change. Just started studying multivariable calc but that's my current grasp on the topic.
• at wouldn't this be the partial derivative with respect to a? his notation got a little confusing.. Like I understand that he means that a is a point on the x axis and b is a point on the y axis, but wouldn't it be more proper to have a formal definition just be in terms of f(x,y) because you can take any point on the x and y axis? Or do any of these details even matter?
• He's showing how to get the derivative at any given point using the formal definition. To get a general df/dx and df/dy equation, it's easier to use the method in the section "Partial derivatives, introduction." You can use the formal definition to find a general derivative equation for most functions, but it is much more tedious, especially with higher polynomial functions. Imagine taking the derivative of f(x,y) = x^5 + x^4y + x^3y^2 + x^2y^3 + x y^4 + y^5 so many limits to take for x and y. It is usually taught so students understand better on what derivatives really are.
• imagine i have taken the partial derivative of f(x)=x^2 with respect to y, what will be the graphical expression of this derivative?
(1 vote)
• You can't take the partial derivative of f(x)=x^2 because it's a one variable function.
• Why do we call the derivative of a multi-variable function as δF/δx and not just f'(x,y) = why is the term "x" used in the generalized 3-d derivative ?
I understand that to find the derivative of F we would have to compute it in terms of both x and y . Thus i find the term "x" in the general derivative of F confusing.
(1 vote)
• If we used f'(x, y) to signify the derivative of f, it would be ambiguous as to what we're taking the derivative with respect to - thus we use the x to say that we're taking the derivative with respect to x, and the same with y. I'm not sure where you're seeing that that's the generalized derivative of f, because the real 'generalized derivative' of a function is the gradient, which you'll see later.
• how can this be a formal definition if it does not consider multiple dimensions?
(1 vote)
• Because a definition can be formal without being completely generalized, i.e., extended to multiple dimensions.
As an example in another subject area, one can define "integer" without defining "number".
• Why partial x 's resulting change(partial f) is in negative direction?
(1 vote)
• He just took it arbitrarily.Depending on the function f ,the change can be in negative or positive direction.
• Do you always need at least two inputs to have a partial derivative? Can you have a partial derivative for one input to the function?
(1 vote)
• Technically, you can. However, think about what it would mean.

∂f(a)/∂x = ?

If you apply the formal definition, you get that:

(∂f(a)/∂x) = lim_{h \to 0} \frac{f(a + h) - f(a)}{h}

So, you just get the definition of df(a)/dx.

So, you ∂f(a)/∂x = df(a)/dx.

Because of that, you don't really see people use partial derivatives for a function of a single input, since it's just equal to the normal derivative of that function. But, it's totally possible lol.
(1 vote)
• Why isn't that you don't typically see the partial derivatives added together to give the full derivative? You see these added with the chain rule applied to multivariable functions (with intermedieate variables) but NOT to simple multivariable functions.
(1 vote)
• as I understand it, adding them won't give us valuable information but will probably give us a useless number because there isn't such a thing as a "full" derivative.

The closest thing to a full derivative is the gradient which allows us to calculate how specific changes in the x and y affect the output f.

The Fx on its own represents how the x variable changes the output of F. (rate of change with respect to x).

The Fy on its own tells us how the y variable changes the output of F as well.

Adding them won't tell us how x and y change the output because x and y can change in different proportions, for example you can change x as twice as much as y, this would make more sense if you think about directional derivatives.

In case of multivariable chain rule, because the variable t changes both x and y at the same time we need to "add" them together to know how t changes f.

I hope I helped.
(1 vote)
• why is h approaching 0 not infinity since it doesn't specifically tell the size of dx?