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

For all values of x and y, $D_1 f(x,y)$ is the value at x, of the derivative of the function $t\mapsto D_1 f(t,y)$. (Note that this is a function from $\mathbb R\rightarrow\mathbb R$). In other words, you can define $g:\mathbb R\rightarrow\mathbb R$ by g(t)=f(t,y), and find $D_1 f(x,y)$ by calculating $g'(x)$, because $g'(x)=D_1 f(x,y)$. 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).

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

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