 # Differentiation and Integration

## Differentiation and Integration Basics

Integration differentiation are two different parts of calculus which deals with the changes. We always differentiate a function with respect to 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 D[f(x)] is called Euler's notation.

$\frac{d(y)}{d(x)}$ 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 a number of 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 its 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_{}^{} 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 which 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.

Question 1: What is the Difference Between Integration and Differentiation?

Answer: From the above discussion, it can be said that differentiation and integration are the reverse processes of each other. Differentiation, as well as integration, are operations which are performed on functions.

If we compare differentiation and integration based on their properties:

1. Both differentiation and integration satisfy the property of linearity, i.e.,k1 and k2 are constants in the above equations.

2. Both differentiation and Integration operations involve limits for their determination.

3. As discussed, both differentiation and integration are inverse processes of each other.

4. The derivative of any function is unique but the integral of every function is not unique. Upon differentiating a polynomial function, the degree of the result is 1 less than the degree of the polynomial function whereas in case of integration the result obtained has a degree which is 1 greater than the degree of the polynomial function.

5. While dealing with derivatives it can be considered derivative at a point whereas, in the integrals, integral of a function over an interval is considered.

6. Geometrically, the derivative of a function describes the rate of change of a quantity with respect to another quantity while indefinite integral represents the family of curves positioned parallel to each other having parallel tangents at the intersection point of every curve of the family with the lines orthogonal to the axis representing the variable of integration.

Question 2: How Integration is Represented?

Solution: The integration of a function f(x) is given by F(x) and it is represented by:

∫ f(x) dx = F(x) + C

where

R.H.S. of the equation indicates the integral of f(x) with respect to x

‘F(x)’ is called anti-derivative or primitive.

‘f(x)’ is called the integrand.

‘dx’ is called the integrating agent.

‘C’ is the constant of integration or arbitrary constant, and

‘x’ is the variable of integration.

Question 3: What are the Differentiation Formulas for Trigonometric Functions?

Solution: The definition of trigonometry is the interaction of angles and triangle faces. We have 6 major ratios here, for example, sine, cosine, tangent, cotangent, secant and cosecant. Based on these ratios, you must have learned basic trigonometric formulas.

The table given below is the equations of trigonometric functions derivatives:

 d/dx(sin x) = cos x                   d/dx(cosec x) = -cosecx cotxd/dx(cos x) = - sin x                 d/dx(sinh x) = cosh xd/dx(tan x) = sec2x                 d/dx(cosh x) = sinh xd/dx(cot x) = -cosec2x            d/dx(tanh x) = sech2xd/dx(sec x) = secx tanx           d/dx(coth x) = -cosech2x