differentiation notations

Leibniz's notation, uses the symbols \[dx\] and \[dy\] to represent infinitesimal increments of \[x\] and \[y\], respectively, just as \[\Delta x\] and \[\Delta y\] represent finite increments of \[x\] and \[y\], respectively.

Consider y as a function of a variable \[x\], or \[y=f(x)\]. If this is the case, then the derivative of \[y\] with respect to \[x\], which later came to be viewed as the limit, \[\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}=\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\], was according to Leibniz the quotient of an infinitesimal increment of \[y\] by an infinitesimal increment of \[x\], or \[\frac{dy}{dx}\].

Suppose \[y\] represents a function \[f\] of variable \[x\], that is \[y=f(x)\]. Then, the derivative of the function \[f\] can be written as \[\frac{dy}{dx}\] or \[\frac{d}{dx}y\] or even \[\frac{df(x)}{dx}\]. High derivatives are written as \[\frac{d^{n}y}{dx^{n}}\]. Which shows that neither \[\frac{d}{dx}\] or \[\frac{dy}{dx}\] is actually a fraction, but rather a type of notation.
In its modern interpretation, the expression ⁠\[\frac{dy}{dx}\]⁠ should not be read as the division of two quantities \[dx\] and \[dy\]; rather, the whole expression should be seen as a single symbol that is shorthand for \[\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}\].

Lagrange's notation, or known as prime notation was in fact invented by Euler and just popularized by the Lagrange. In Lagrange's notation, a prime mark denotes a derivative. If \[f\] is a function, then it's derivative evaluated at \[x\] is written as \[f^{\prime}(x)\]. Higher derivatives are indicated using additional prime marks, as in \[f^{\prime\prime}(x)\] for the second derivative and \[f^{\prime\prime\prime}(x)\] for the third derivative. Even higher derivatives are usually represented as \[f^{(n)}(x)\].

Isaac Newton's notation for differentiation, also known as dot notation, places a dot over the dependent variable. If \[x\] is a function \[t\], then the derivative of \[x\] with respect to \[t\] is written as \[\dot{x}\]. Higher derivatives are represented with more dots, i.e. \[\ddot{x}\] and \[\dddot{x}\].

• \[dx\] represents an infinitesimal change in the variable \[x\].
  
• \[\Delta x\] represents a finite change in variable \[x\]. In the limit definition of the derivative, we have \[\frac{dy}{dx}=\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}\], where \[\Delta x\] is the finite difference that approaches zero in the limit process.
  
• \[\delta x\] represents small changes in variable \[x\], but not infinitesimally small like \[dx\].
  
• \[\partial x\] represents an infinitesimal change in the variable \[x\] when considering functions of multiple variables. In the context of a function \[f(x,y,z,\dots)\], the partial derivative of \[f\] with respect to \[x\] is denoted by \[\frac{\partial f}{\partial x}\]. This represents how \[f\] changes as \[x\] changes while \[y,z,\dots\] are held constant. \[\frac{\partial f}{\partial x}=\lim_{\Delta x\to0}\frac{f(x+\Delta x,y,z,\dots)-f(x,y,z,\dots)}{\Delta x}\].