Differentiation and Integration

Integration differentiation is two different parts of calculus that deal with the changes. We always differentiate a function from a variable because the change is always relative. Integration is almost the reverse of differentiation and it is divided into two - indefinite integration and definite integration. 

What is Differentiation?

Differentiation can be defined as a derivative of independent variable value and can be used to calculate features in an independent variable per unit modification.

Let,

y = f(x), be a function of x.

Then, the rate of change of “y” per unit change in “x” is given by,

\[\frac{dy}{dx}\]

If the function f(x) undergoes an infinitesimal change of h near to any point x, then the derivative of the function is depicted as

\[\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\]

When a function is depicted as y = f(x),

Then the derivative is depicted by the following notations:

D(y) or Df(x)

F(x) is called Euler's notation.

\[\frac{dy}{dx}\] is known as Leibniz’s notation.

F(x) is known as Lagrange’s notation.

Differentiation is the method of evaluating a function's derivative at any time.

Differentiation Rules:

To understand differentiation and integration formulas, we first need to understand the rules. Some of the fundamental rules for differentiation are given below:

Sum or Difference Rule:

When the function is the sum or difference of two functions, the derivative is the sum or difference of derivative of each function, i.e.

If f(x) = u(x) ± v(x), then f’(x) = u’(x) ± v’(x)

Product Rule:

When f(x) is the sum of two u(x) and v(x) functions, it is the function derivative,

If f(x) = u(x) x v(x),

Then f’(x) = u’(x) x v(x) + u(x) x v’(x)

Quotient Rule:

If the function f(x) is in the form of two functions

\[\frac{u(x)}{v(x)}\], the derivative of the function can be expressed as:

If \[f(x)=\frac{u(x)}{v(x)}\],

Then \[f'(x)=\frac{u'(x)\times v(x)-u(x)\times v'(x)}{[v(x)]^2}\]

Chain Rule:

If y = f(x) = g(u), 

And if u = h(x)

Then, \[\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}\].

Here is a differentiation theorem collection for students so that they can turn to them to solve differential equations related problems. Higher-level mathematics is one of the most important topics. The general depiction of the derivative can be expressed as d/dx.

This list of formulas contains derivatives for constant, polynomials, trigonometric functions, logarithmic functions, hyperbolic, trigonometric inverse functions, exponential, etc. There are many examples and issues in class 12 courses, which can be easily addressed by students.

What is Integration?

Calculus consists of two main operations and Integration is inverse operation and differentiation is one of them. Given a function f(x) of a real variable x and an interval a, b of the real line can be represented as follows:

\[\int_{a}^{b} f(x) dx\]

It can be explained informally as the signed area of the region in the xy-plane which is bounded by the graph of f(x), the vertical lines(x = a and x = b) and the x-axis. The area below the x-axis always subtracts from the total whereas the area above the x-axis adds to the total. 

The inverse of the operation of differentiation is the operation of integration, up to an additive constant. Thus, the term integral also means the related notion of the anti-derivative, a function f(x) whose derivative is the given function. This is called indefinite integral and is written as:

\[F(x)=\int f(x) dx\]

Definite integrals relate differentiation with the definite integral: if f(x) is a continuous real-valued function which is defined on a closed interval a,b. Therefore, the definite integral of f over that interval is shown by:

\[\int_{a}^{b}f(x)dx=[F(x)]_{a}^{b}=F(b)-F(a)\]

Isaac Newton and Gottfried Wilhelm Leibniz formulated the principles of integration, independently in the late 17th century. Integral was thought to be an infinite sum of rectangles having infinitesimal width. A rigorous mathematical definition of integrals came from another Mathematician named Bernhard Riemann. The limiting procedure approximates the area of a curvilinear region only by breaking the region into thin vertical slabs. There are two types of integral:

1. A line integral defines functions of two or more variables, where the interval of integration a,b is replaced by a curve that connects the two endpoints.

2. A surface integral is an integral where the curve is replaced by a piece of a surface in 3D space.

Real-Life Examples of Differentiation and Integration provided by Vedantu

The idea of differentiation is very simple. If we have a quantity, e.g., a number, which has its natural order, and we ask how it changes over time (its rate of change), we have a function, and the derivative of that function is the rate at which the quantity changes. For example, we may have the number of people in a room at a moment in time, and we ask how that number has changed over time. In this case, the number of people, the rate at which they enter and leave the room, is a function (or mapping) from time to time. 

The rate at which people enter and leave the room changes as time progresses, so the derivative of that function is the function that tells us how many people change from entering the room to leaving the room. Since all the information about how many people change from entering to leaving is contained in the derivative function, we say that the derivative is the function that gives us the rate at which the number changes.

A similar example may be found in describing the rate of flow of gas; the flow rate may be a function from time to a number. The derivative function then tells us how the gas flow rate changes over time.

The idea that the rate at which a function change is its derivative is the foundation of calculus. This idea was developed in the 18th and 19th centuries and was formalized by many mathematicians beginning in the 19th century. It is commonly taught in many high schools.

To understand the idea of integration, first, let’s consider how to find the area of a square. If we find the area of a square (which has its natural order), and ask how much it changes over time, then the answer is simple: A square of width 5 changes the area by 25% when time changes by 25%, and by -25% when time changes by -25%. This answer gives the rate of change of the area of the square and can be written as

Note that the area of the square, which is 25, is a constant. What it means to be a constant is explained here. It means that if we were to draw the graph of a constant function (a function whose value does not change over time), the function would be a line on which the points have the same horizontal coordinate.

Now imagine we are asked to find the area of the square if the size of the square changes at a rate of 5 every 15 minutes, i.e., if the size of the square changes from a width of 5 to a width of 6 every 15 minutes.

We have to figure out how the area of the square changes over time. This would involve solving the system of two simultaneous equations. That is: the area of the square is the product of the width and the length, and at the time 15 minutes have passed, the width is 6 and the length is 10. We know that the length of a square is the width of the square times its height, and we know that the length of a square is 10. So the product of these is the area of a square. Therefore, the area of the square is 10,000 (remember, the area of a square is the width times its height).

The rate of change in the area is also 10,000.

There are a few important things to note. In this case, the constant is 5. What we meant by that earlier was that the area does not change. So the rate of change of the area is zero, or in this case, 10,000. We have seen that the area of a square does not change. Therefore, the function is not increasing.

Some Common Mistakes

Differentiation and Integration Basics in mathematics have the basic foundation in algebra. With a strong background in algebra, one can get the basics of differentiation and integration in simple steps. This article will highlight some of the common mistakes that appear in algebra tests so that students can identify and rectify the mistakes.

1. Identifying the type of a problem

It is important to know which type of problem is given to one so that one can focus on it. When the problem is given, if there is any symbol like x, it means that the variable a has been defined. In the case of y, it implies that the variable b is defined. Sometimes the problem contains three variables, and in this case, you have to look out for the coefficient of a particular variable and the coefficient of the remaining two variables. If the problem has two variables, one with its coefficient and the other with its coefficient, then we can differentiate the other variable for it. If the problem is of four variables, then it will have the coefficient of each variable. In the case of a function, variable a will have a coefficient, and invariable a, another variable b will be defined.

2. Identifying the coefficient of a variable

You can determine the coefficient of a variable by using the basic equation 2abc. For example, if the problem states, solve for a.

3. Identifying the type of an equation

You can determine the type of an equation by using the basic equation 2abc. The following examples show how the basic equation is used to determine the type of an equation. The type of an equation will be one of the following:

• linear

• quadratic

• cubic

• rational

• exponential

4. Identifying a given function

There are two different types of a given function: the absolute value and the inverse function. The absolute value of a function is usually defined as f(x) = |g(x)|, where f(x) is the absolute value of g(x), and g(x) is the function. An inverse function is usually defined as f(x) = g(x). There are three ways to define a given function as either an absolute value or an inverse function. One way is to take the absolute value of either g(x) or f(x). Another way is to change the sign of g(x) or f(x). The third way is to change the sign of both g(x) and f(x).

5. Identifying the type of a graph

Graphs are useful for showing the relationship between two variables. There are four types of graphs. The first is a linear graph that is often found in the basic equation 2abc. Another kind of graph is a quadratic graph, which is used when the basic equation 2abc involves the square of a variable. The third is a cubic graph that shows the relation between two variables when the basic equation is cubic in one variable. The fourth kind of graph is the rational graph, which shows the relation between two variables when the basic equation is rational. The following graph is of a quadratic function, which involves the square of the variable.