#### • Class 11 Physics Demo

Explore Related Concepts

# Differentiation and Integration Formulas

Mathematics is a very interesting subject. There are many formulae that are used in mathematics to solve problems. The differentiation and integration formula is one among them.
Both these are used in the field of calculus. They are very important part of calculus and form the backbone of calculus.
To understand the concept of differentiation one must also understand the concept of function. There are inputs to the function.
These inputs might change over a period of time.

So, differentiation in Calculus can be used to find the derivative of a function.
This derivative can be used as a measure of knowing how much the function changes with the changes in the inputs that are given to the function. So, one can use the concept of differentiation to find the derivative.  Various functions are inputs to the differentiation formula.
There is a list of formulae that could be used for the purpose of finding the differentiation of a function.
The differentiation of an exponential raised to the power of ‘x’ is the same exponential raised to the power of ‘x’. This is one of the formulae.

There are several formulae and these depend on the function that is given as the input.
Integration on the other hand can be thought of the inverse process of differentiation.
There can be both definite and indefinite integrals. For the definite integrals limits are defined but for the indefinite integrals no limits are defined.

Basically when the limits are used the integration is used to represent the area that is bounded by the function, over which integration is performed. This area is present in the x-y plane. The differentiation and integration formula must be studied in order to solve the problems based on differentiation and integration. Both these tools are very important tools of Calculus and have various applications in different fields.

When the final limits of the integration are to be applied then the value of the upper limit is put into the function first and then the value of the function is found out. Similarly the value of the lower limit is put into the function and the value of the function is found out.
Now the difference between the value obtained by putting the upper limit value in the function and the value obtained by putting the lower limit value in the function is found out. This is the value of the finite integral of the function over which integration is performed.

In integration there are concepts of line integral and surface integral. The line integral can be performed over functions that can have two or three variables.
A variable is one whose value changes over and is not fixed or constant.
The surface integral can be performed over a piece of space that is present in the three dimensional space.
This is the basic difference between the two types. This has to be kept in mind while performing the operation.

The differentiation and integration formula must be learnt thoroughly and only then they can be effectively used to solve the problems.
Sometimes double or triple integrals can also be performed. They are a bit more complex than the simple integrals but the procedure remains the same. The differentiation of a constant number is zero.
This is one of the formulae of differentiation.

An example of differentiation can be given from the field of science. If an object is moving, the rate of change of its position is found with respect to time then the value obtained is nothing but the velocity of the moving object.
So, this is one of the simple applications.

From Wikipedia

Differential calculus

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus.

The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.

Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.

Differentiation has applications to all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. Newton's second law of motion states that the derivative of the momentum of a body equals the force applied to the body. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories. By applying game theory, differentiation can provide best strategies for competing corporations.

Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra.

## The derivative

Suppose that x and y are real numbers and that y is a function of x, that is, for every value of x, we can determine the value of y. This relationship is written as: y = f(x). Where f(x) is the equation for a straight line, y = mx + b, where m and b are real numbers that determine the locus of the line in Cartesian coordinates. m is called the slope and is given by:

m={\mbox{change in } y \over \mbox{change in } x} = {\Delta y \over{\Delta x}},

where the symbol Î” (the uppercase form of the Greek letter Delta) is an abbreviation for "change in". It follows that Î”y = mÎ”x.

In linear functions the derivative of f at the point x is the best possible approximation to the idea of the slope of f at the point x. It is usually denoted f'(x) or dy/dx. Together with the value of f at x, the derivative of f determines the best linear approximation, orlinearization, of f near the point x. This latter property is usually taken as the definition of the derivative. Derivatives cannot be calculated in nonlinear functions because they do not have a well-defined slope.

A closely related notion is the differential of a function.

When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f' at x. Because the source and target of f are one-dimensional, the derivative of f is a real number. If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once. Taking the best linear approximation in a single direction determines a partial derivative, which is usually denoted âˆ‚y/âˆ‚x. The linearization of f in all directions at once is called thetotal derivative. It is alinear transformation, and it determines the hyperplane that most closely approximates the graph of f. This hyperplane is called the osculating hyperplane; it is conceptually the same idea as taking tangent lines in all directions at once.

Question:Can anyone explain me what is the use of integration and differentiation. i have learnt about it. But what is its use

Answers:Let us get into this with all interest. Yes integration is a reverse process of differentiation. First let us put the question why do we need such a mathematical process called integration. Actually integration is nothing but the summing up of a lot, some million and million items. Let me explain how. Suppose you want to find the volume of a cone of radius r and height h. Let the cone be seated such that its vertex get coincided with the origin and its height be along the x -axis. In case of a cylinder the radius will be the same at all heights and so if you consider a small part both the sides of that part will have the same radius and so no problem in finding the volume. But in the case of the cone, as we move away from the origin along x-axis the radius of the cone will be gradually increasing. So as you consider a slice any where in the cone both the sides of the slice will not definitely have same radius. Is that ok? Now calculus comes into play. You choose a small slice in such a way that both the sides of the slice would have almost the same radius. It is possible only when you have a slice of negligible thickness. Such negligible thickness is denoted mathematically denoted as dx which means delta x tending to zero. Once again note down the statement: delta x tending to zero. This means delta x is very so near to zero but not zero. With such....... thickness, both sides of the slice would have the same radius.Let the slice of thickness dx be chosen at a distance x from the origin and let the radius of the slice be y at that position. Then the volume of our slice will be pi y^2 dx. Now imagine! Such slices, innumerable in number, can be got in the cone right moving from the origin and extending upto the total length (height) h of the cone. So we must collect all such slices and add their volumes to get the actual or total volume of the cone. So we integrate the term pi y^2 dx within the limits of x ie 0 to h. Now it becomes more essential to replace y interms of x. How can we do this? By using similar triangle concept, the ratio of the corresponding sides will be the same. So y/x = r/h From this we can easily have y as (r/h) x. Now replacing y we get the expression to be integrated (usually named as integrand) as pi (r/h)^2 x^2 dx. The formula for integral of x^n dx is given as x^(n+1) /(n+1). So following this we get the integral value as pi(r/h)^2 (x^3 /3). Next important step is supplying limit of x. First upper limit h. This would give a value of pi(r/h)^2 (h^3 /3) With lower limit 0, the value would become 0 Now the difference between these two values will be the required volume of the cone. That comes to be 1/3 pi r^2 h. (cancelling h^2) So interesting! See how much helpful the technique of integration in finding the volume of the cone! By differentiation, we chop things into finer and by integration we collect all such finer. Hope you have got a gist of the tremendous usage of the branch of mathematics, named as calculus. Best wishes to get assimilated soon.

Question:Hey, I just registered for this math course in my university and it involves a lot of integration and differentiation. I have never learned about these before, my math background is more based on using graphic calculators and such. Anyway, I want to know if there are some good places to quickly learn the basics of integration and differentiation, their meanings, formulas i need to have memorized, etc. Thanks!

Answers:there are 2 great ways. One of them is not the following two links that covers the rules Derivatives: http://www.alcyone.com/max/reference/maths/derivatives.html Integrals: http://www.alcyone.com/max/reference/maths/integrals.html The next one is my most favorite, call Open Course-Ware. It's the program they have at MIT that gives the general public access to all sorts of stuff on their class, EVEN VIDEOS OF THE LECTURES! http://ocw.mit.edu/courses/mathematics/ ^^That's the math page. Use it wisely!

Question:I want to learn Differentiation & integration in maths Please give me idea about it.

Answers:The best way is to learn these is you must go to ur teacher and request them to give you extra time and learn formulae by practicing alot and discuss al the difficulties with ur teachers and friends

Question:Do I have to get used with the approximation formula?

Answers:jus buy a mini computer type of calculator and that should replace some functions of your brains and do all the standard computations. You don't have to be a calculator yourself.