Difference in use between $d$, $\partial$, $\operatorname d$, $\varDelta$ and $D$ for derivatives.

Question

While reading different sources on implicit differentiation (and thereafter differentiation in general), I came across many different "d's" being used for (or similar to) the familiar $$\frac{dy}{dx}$$ The variant $d$ appears in the majority of cases, and also with integrals:

$$\int x~dx$$

Now, it appears like the convention on Wikipedia is to use $d$ or $\operatorname d$ for "normal" and $\partial$ for partial (which is also its name in Tex) derivatives. Is there an additional difference between $d$ and $\operatorname d$ (except the amount of characters needed), and are there any general conventions?

Answer 1

I regard $d$ and $\rm d$ as the same. In general, I think of it like this:

• If $f: \Bbb R \to \Bbb R$, the derivative at $x$ is $f'(x)$ or $\frac{{\rm d}f}{{\rm d}x}(x)$;
• If $f: \Bbb R^n \to \Bbb R$, the $k-$th partial derivative at $x$ is $\frac{\partial f}{\partial x^k}(x)$, or $D_kf(x)$, or $D_{x^k}f(x)$, or $\partial_kf(x)$ or $\partial_{x^k}f(x)$;
• If $f: \Bbb R^n \to \Bbb R^m$, the total derivative (Fréchet derivative, if you like it) at $x$ is $Df(x)$. In particular, we can call the partial derivative $\frac{\partial f}{\partial x^k}(x)$, which will be a vector whose components are the partial derivatives of the components, following the above item. If $f: \Bbb R^n \times \Bbb R^m \to \Bbb R^k$ we can also use $D_xf(x,y)$ and $D_yf(x,y)$ to denote blocks of $Df(x,y)$. This is useful sometimes when doing things with the inverse/implicit function theorems.
• If $f: \Bbb R^n \to \Bbb R^m$, the differential of $f$ at $x$ is the map ${\rm d}f_x: T_x\Bbb R^n \to T_{f(x)}\Bbb R^m$ given by ${\rm d}f_x(v_x) = (Df(x)(v))_{f(x)}$. Some people write ${\rm d}_xf$ instead of ${\rm d}f_x$.
• The exterior derivative of $k$-form $\omega$ is denoted simply by ${\rm d}\omega$.

Answer 2

• $D_x$ is the same thing as $\frac{d}{dx}$, it's just a different notation inveted by Euler. The capital $D$ of course stands for derivative, and derivative is with respect to the "base" of $D$, for example $D_t$ is derivative in respect to $t$.

• $\frac{\partial}{\partial x}$ is a partial derivative used in multi-variable calculus. In this case, it is with respect to $x$, so other unknown variables other than $x$ would be set constant.

• $\frac{\text{d}}{\text{d}x}$ is also not different from $\frac{d}{dx}$. Both d and $d$ are the same letters and I don't know why slanted $d$ would differ in any way from normal d. It is also used in integrals, as you said as d$x$ or $dx$, which is the differential form, and it is there so you know with respect to which variable, should you integrate.

• $\Delta x$ notes change in $x$, which is used in the definition of derivative for a point. The definition for derivative is: $$\lim_{\Delta x\to 0}f'(x)=\frac{f(x+ \Delta x)-f(x)}{\Delta x}$$ where, as I stated $\Delta x$ is change in $x$, which in calculation of derivatives, approaches $0$.