hi..........may i ask you something?????????

Question

what is differentiation and integration in calculus?????????????????????????

Answer 1

Differentiation
Expresses the rate at which a quantity, y, changes with respect to the change in another quantity, x, on which it has a functional relationship. Using the symbol Δ (Delta) to refer to change in a quantity, this rate is defined as a limit of difference quotients which means the limit as Δx approaches 0. In Leibniz's notation for derivatives, the derivative of y with respect to x is written suggesting the ratio of two infinitesimal quantities. The above expression is pronounced in various ways such as "d y by d x" or "d y over d x". The form "d y d x" is also used conversationally, although it may be confused with the notation for element of area.

Modern mathematicians do not bother with "dependent quantities", but simply state that differentiation is a mathematical operation on functions. One precise way to define the derivative is as a limit [2]:


A function is differentiable at a point x if the above limit exists (as a finite real number) at that point. A function is differentiable on an interval if it is differentiable at every point within the interval.

As an alternative, the development of nonstandard analysis in the 20th century showed that Leibniz's original idea of the derivative as a ratio of infinitesimals can be made as rigorous as the formulation in terms of limits.

If a function is not continuous at a point, then there is no tangent line and the function is not differentiable at that point. However, even if a function is continuous at a point, it may not be differentiable there. For example, the function y = |x| is continuous at x = 0, but it is not differentiable there, due to the fact that the limit in the above definition does not exist (the limit from the right is 1 while the limit from the left is −1). Graphically, we see this as a "kink" in the graph at x = 0. Thus, differentiability implies continuity, but not vice versa. One famous example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function.

The derivative of a function f at x is a quantity which varies if x varies. The derivative is therefore itself a function of x; there are several notations for this function, but f' is common.

The derivative of a derivative, if it exists, is called a second derivative. Similarly, the derivative of a second derivative is a third derivative, and so on. A function may have zero, a finite number, or an infinite number of derivatives.



Integration
In calculus, the integral of a function is an extension of the concept of a sum. The process of finding integrals is called integration. The process is usually used to find a measure of totality such as area, volume, mass, displacement, etc., when its distribution or rate of change with respect to some other quantity (position, time, etc.) is specified. There are several distinct definitions of integration, with different technical underpinnings. They are, however, compatible; any two different ways of integrating a function will give the same result when they are both defined.

The term "integral" may also refer to antiderivatives. Though they are closely related through the fundamental theorem of calculus, the two notions are conceptually distinct. When one wants to clarify this distinction, an antiderivative is referred to as an indefinite integral (a function), while the integrals discussed in this article are termed definite integrals.

The integral of a real-valued function f of one real variable x on the interval [a, b] is equal to the signed area bounded by the lines x = a, x = b, the x-axis, and the curve defined by the graph of f. This is formalized by the simplest definition of the integral, the Riemann definition, which provides a method for calculating this area using the concept of limit by dividing the area into successively thinner rectangular strips and taking the sum of their areas then the integral of f between a and b is a measure of S. In intuitive terms, integration associates a number with S that gives an idea about the 'size' of the set (but this is distinct from its Cardinality or order). This leads to the second, more powerful definition of the integral, the Lebesgue integral.

Leibniz introduced the standard long s notation for the integral. The integral of the previous paragraph would be written . The sign represents integration, a and b are the endpoints of the interval, f(x) is the function we are integrating known as the integrand, and dx is a notation for the variable of integration. Historically, dx represented an infinitesimal quantity, and the long s stood for "sum". However, modern theories of integration are built from different foundations, and the notation should no longer be thought of as a sum except in the most informal sense. Now, the dx represents a differential form.

Answer 2

Differentiation is simplifying a complex equation (similar to division)

Integration is a way to get higher and complex values (similar to addition or multiplication)

Answer 3

Deravative is essentially the slope of a curve.
limit of (f(x+d)-f(x))/d

Integral is the area of the curve. Limit of summing the rectangles under it as the rectangles get smaller.

I forgot intuitively why one is the opposite of the other.

Answer 4

I really want to ask something and i think the only way to get points here is to answer questions. I know right? ******* stupid! so yeah, this is my answer, i better get some ******* points because im annoyed >:(