What is differential and integration classified into? A function or operation?

Question

Operations are simple process we do to numbers to add and multiply and, subtract and divide. A function is a combination of numbers and, operator/operations and, some variables, this has no limit how much number or variable one put inside.

My question is, what is differential and integration classified into? Seems like a process so it's an operation but not just number but other operators also.

Answer 1

Integration and differentiation are both functions of functions. You put in a function, you get out a function. Often functions of functions are called operators as well.

Concerning making up names in maths in general: There is no governing body that decides on how to call things in maths. Names are more fluid and therefore anyone who writes a book or holds a lecture needs to painstakingly define every object they talk about, before working with them.

Answer 2

In higher mathematics, there are no "processes" in the sense you mean. There are objects, sets of objects, and functions between sets of objects (and in the standard foundation of math, these are actually all sets, but it's still useful to categorize them differently depending on their purpose). Addition is a function. It takes two objects (numbers) from a set (the reals, for instance) as input, and gives an object from the same set as an output. In this sense, $1+2$ is not a process, it is an object, a number. Specifically, it is the output of addition if you give it the numbers $1$ and $2$ as input, so $3$. $1+2$ does not "become" $3$ or anything like that. It is $3$.

Derivatives and integrals are objects. The derivative of $f(x)=x^2$ at $x=1$ is the object $2$ (a number). The integral $\int_0^1 x\mathrm dx$ is the object $\frac12$ (a number).

However, derivative functions and integral functions are functions, as the name implies. The derivative function of $f(x)=x^2$ takes a number $x$ as input, and gives the derivative $f'(x)$ (which is $2x$) as an output. The integral function $J_0(x)=\int_0^x t\mathrm dt$ takes a number $x$ as input and gives the integral written above as an output (which turns out to be $\frac12 x^2$.

Or you can go even further and consider the function which assigns to each differentiable function its derivative function. This is also a function.

Answer 3

It depends how you are defining things, in the below expression $\frac{d}{dx}$ is the operator acting on the function $f(x)$ and the output $f'(x)$ is also a function: $$\frac{d}{dx}f(x)=f'(x)$$ Also note the use of the word mapping.