I think rather d/dx is the operator. You apply it to an expression to bind free occurrences of x in that expression. For example, dx²/dx is best understood as d/dx (x²). The notation would be clear if you implement calculus in a program.
d is just an infinitesimally small delta. So dy/dx is literally just lim (∆ ->0) ∆y/∆x.
which is the same as lim (x_1 -> x_0) [f(x_0) - f(x_1)] / [x_0 - x_1].
Note: ∆ ->0 isn’t standard notation. But writing ∆x ->0 requires another step of thinking: y = f(x) therefore ∆y = ∆f(x) = f(x + ∆x) - f(x) so you only need ∆x approaching zero. But I prefer thinking d = lim (∆ -> 0) ∆.
I think rather
d/dxis the operator. You apply it to an expression to bind free occurrences ofxin that expression. For example,dx²/dxis best understood asd/dx (x²). The notation would be clear if you implement calculus in a program.I just think of the definition of a derivative.
dis just an infinitesimally small delta. Sody/dxis literally justlim (∆ -> 0) ∆y/∆x. which is the same aslim (x_1 -> x_0) [f(x_0) - f(x_1)] / [x_0 - x_1].Note:
∆ -> 0isn’t standard notation. But writing∆x -> 0requires another step of thinking:y = f(x)therefore∆y = ∆f(x) = f(x + ∆x) - f(x)so you only need∆xapproaching zero. But I prefer thinkingd = lim (∆ -> 0) ∆.If not fraction, why fraction shaped?
If you use exterior calculus notation, with d = exterior derivative, everything makes so much more sense