Regular Derivative and A Partial Derivative

[bmed90]

Can someone please explain to me the difference between a regular derivative and a partial derivative?

 

[Bacle]

I think if you try to define each of the two, you will see the difference. Let us know otherwise.

 

[felper]

If you take both derivatives on a single variable function, they will be the same. What's the definition of the "regular" derivate of a several variables function? Check this

https://www.physicsforums.com/showthread.php?t=277979

 

[Bacle]

I didn't mean too be soo abrupt; it is just difficult to answer these questions without more context. If you would tell us some more of what is on your mind?

 

[Fredrik]

Suppose that [itex]f:\mathbb R^2\rightarrow\mathbb R[/itex]. The partial derivative of f with respect to the ith variable is a function from [itex]\mathbb R^2\rightarrow\mathbb R[/itex]. I like to denote it by [itex]D_i f[/itex] or [itex]f_{,i}[/itex]. So I would denote the value at (x,y) of the ith partial derivative of f by [itex]D_i f(x,y)[/itex] or [itex]f_{,i}(x,y)[/itex]. I'll stick to the D notation in this post.

For all values of x and y, [itex]D_1 f(x,y)[/itex] is the value at x, of the derivative of the function [itex]t\mapsto D_1 f(t,y)[/itex]. (Note that this is a function from [itex]\mathbb R\rightarrow\mathbb R[/itex]). In other words, you can define [itex]g:\mathbb R\rightarrow\mathbb R[/itex] by g(t)=f(t,y), and find [itex]D_1 f(x,y)[/itex] by calculating [itex]g'(x)[/itex], because [itex]g'(x)=D_1 f(x,y)[/itex]. We can obviously make a similar comment about partial derivatives with respect to the second variable. So every calculation of the value of a partial derivative at a point in its domain is a calculation of the value of an ordinary derivative at a point in its domain. This is a fact that I don't think is emphasized often enough.

Example: If you're asked to compute the partial derivative of xy² with respect to x, it can be interpreted as: Let f be the function defined by f(t)=ty² for all t. Find f'(x) (i.e. the derivative of f, evaluated at x). If you're asked to compute the partial derivative of xy² with respect to y, it can be interpreted as: Let g be the function defined by g(t)=xt² for all t. Find g'(y) (i.e. the derivative of g, evaluated at y).

[tex]\begin{align}&\frac{\partial}{\partial x}xy^2=(t\mapsto ty^2)'(x)\\ &\frac{\partial}{\partial y}xy^2=(t\mapsto xt^2)'(y)\end{align}[/tex]
There is really no difference between the expressions [tex]\frac{d}{d x}xy^2[/tex] and [tex]\frac{\partial}{\partial x}xy^2[/tex] for example. The latter is defined to mean [tex]D_1\big((s,t)\mapsto st^2\big)(x,y),[/tex] but this is (by definition of [itex]D_1[/itex]) equal to [tex](s\mapsto sy^2)'(x),[/tex] which is what the former is defined to mean.

So one valid way of thinking of expressions of the form [tex]\frac{\partial}{\partial x}\big(\text{Something that involves x and at least one more variable}\big)[/tex] is that the partial derivative notation is just telling you which function from [itex]\mathbb R[/itex] into [itex]\mathbb R[/itex] to take an ordinary derivative of, and at what point in the domain to evaluate that derivative.