Differentiation and Integration Overview
Differentiation and integration are essential areas of calculus, and the differentiation and integration formulas are mutually exclusive. Calculus' two main principles are differentiation and integration. Differentiation is used to investigate the small change in one quantity in relation to the unit change in another. Integration, on the other hand, is used to integrate tiny and discontinuous data that cannot be added separately and represented in a single number, so an integral is also known as the antiderivative. The rate of change in speed with respect to time (i.e. velocity) is a real-life example of differentiation, and the most significant example of integration is determining the area between the curves for large-scale companies. Geometrically, the differentiation and integration formulas are employed to calculate the slope and area under a curve.
What is Calculus?
Calculus is one of the most important mathematical applications used in the world today to tackle a variety of problems. It is widely used in scientific research, economic studies, finance, and engineering, among other fields that play an important part in an individual's life. Calculus' basics for studying change are integration and differentiation. Many individuals, even students, and experts have been unable to distinguish between distinction and integration.
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What are Differentiation and Integration?
What exactly is Differentiation?
Differentiation in mathematics is defined as the derivation of any function using an independent variable. It is also known as the function per unit change. It is represented as dy/dx or y', where x is the independent variable and y differ from x. If any function has a very tiny change 'k' near to x, the derivative is lim k0 f(x+k)-f(x) / k.
Example for Differentiation
If y = f(x) is a function of x, the rate of change of y per unit change in x is denoted by-
|
dy/dx |
Types of Differentiation
Differentiation rules are classified into four types-
- Chain Rule- This rule governs the differentiation of a mixed function. The objective is to first solve the outside function, then the inside function. Tan(k²), for example, is a composite function since it can be represented as m(n(x)) for m(x) = tan(k) and n(x)=k².
- Product Rule- d/dx (pq) = q * dp/dx + p * dq/dx
- Quotient Rule- d/dx {p(x) / q(x)} = [q(x) * p'(x) – p(x) * q'(x)]/ p(x)2
- Sum and Difference Rule- F(X) () F(Y) = F'(X) () F'(Y)
What exactly is Integration?
Integration is the process of joining pieces to produce a whole. In integral calculus, we obtain a function whose differential is supplied. Differentiation is thus the inverse of integration. Integration is used to define and compute the size of the region encompassed by the graph of functions. The area of the polygon etched in the curved shape is approximated by tracking its integrand. This" sign represents the integration of a function.
Example for Integration
For a function f(x) and a closed interval [a, b] on the real line, the definite integral is denoted by-
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∫ba f(x) dx |
Types of Integrals
Integrals are classified into two types, which are as follows-
- Indefinite Integrals- As the name implies, an indefinite integral is a function's integral when there is no limit to integration. It contains an arbitrary constant denoted by the symbol 'C'.
- Definite Integrals- A definite integral is a function integral with integration constraints. The bounds of the integration interval are two values. The lower value represents the lower limit, while the larger value represents the upper limit. It contains no integration constant.
The notions of integration were created by the famous mathematician Leibniz.
Fundamental Integration Methods
There are four fundamental integration methods, which are as follows-
- Integration by Decomposition Method
- Integration by Parts
- Integration by Substitution
- Integration Using Partial Fractions
Read more about Who Invented Math and Who is the Father of Mathematics.
Differentiation and Integration Derivations
Differentiation Derivations
If the derivative f'(a) occurs at every point in its domain, a function f in x is said to be differentiable at the point x = a. A function f'(x) derivative is given by-
To be differentiable at any point x = an in its domain, a function must be continuous at that point, but vice versa is not always true. The presence of limits defines the domain of f'(x).
If y = f(x) is a function in x, then dy/dx is the derivative of f(x). This is referred to as y's derivative with regard to x.
Similarly, at x = a, the derivative of a function f(x) is given by-
d/dx (f(x))|ₐ
The derivative of a function f(x) represents the function's rate of change with respect to x at a point 'a' in its domain.
We can determine the function f if we know the derivative of the function, f', which is differentiable in its domain. f is the anti-derivative or primitive function f' in integral calculus. Anti-differentiation or integration is the process of computing the anti-derivative.
Integration Derivations
Integration is a technique for calculating definite and indefinite integrals. F(x) denotes the integration of a function f(x) and is represented by-
where R.H.S. of the equation indicates the integral of f(x) with respect to x F(x),which 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.
x is the variable of integration.
Differentiation and Integration Formula
The following table gives formulas for differentiation and integration-
|
Differentiation Formulas |
Integration Formulas |
|
d/dx (a) = 0 where a is constant |
∫ 1 dx = x+C |
|
d/dx (x) = 1 |
∫ a dx = ax + C |
|
d/dx(xⁿ) = nxⁿ⁻¹ |
∫ xⁿ dx = (xⁿ⁺¹/n+1) + C |
|
d/dx sin x = cos x |
∫ sin x dx = -cos x + C |
|
d/dx cos x = -sin x |
∫ cos x dx = sin x + C |
|
d/dx tan x = sec² x |
∫ sec² x dx = tan x + C |
|
d/dx ln x = 1/x |
∫ (1/x) dx = ln x + C |
|
d/dx eˣ = eˣ |
∫ eˣ dx = eˣ + C |
Differentiation and Integration Properties
- Differentiation and integration are both activities that need limitations to be determined.
- As previously stated, differentiation and integration are inverse processes.
- The derivative of any function is one-of-a-kind, whereas the integral of any function is not. A constant can separate two integrals of the same function.
- When a polynomial function is differentiated, the degree of the result is one less than the degree of the polynomial function, but when a polynomial function is integrated, the degree of the result is one larger than the degree of the polynomial function.
- While dealing with derivatives, we can consider the derivative at a single point, whereas integrals consider the integral of a function across an interval.
- Geometrically, a function's derivative describes the rate of change of one quantity with respect to another, whereas an indefinite integral represents a family of parallel curves with parallel tangents at the intersection points of each curve in the family, with the lines orthogonal to the axis representing the variable of integration.
Integration Application
Integration may be defined as a mathematical method of adding slices to discover the total. This integration technique may be used to quickly obtain areas, volumes, median points, and other important data.
Differentiation and application of the Integral Sign
Differentiation under the integral sign is an algebraic procedure in calculus used to evaluate specific integrals. This procedure allows us to shift the order of integration and differentiation when the function being integrated is relatively loose. Its simplified version is known as the Leibniz integral rule, and in this version, differentiation under the integral sign models the resultant equation legitimately according to the following formula-
|
ddt ∫baf(x,t) dx = ∫ba∂tf(x,t) dx |
This method may be used to solve numerous integrals that would otherwise be impossible or require a far more sophisticated approach. Differentiation according to the integral sign rule can also be utilized to evaluate some uncommon definite integrals.
Similarities Between Differentiation and Integration
- Both differentiation and integration satisfy the following commonalities and common formulas-
- They meet the linearity requirement, that is, d(f(x) ± g(x))/dx = d(f(x))/dx ± d(g(x))/dx and ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx.
- Differentiation and integration are antithetical processes.
- They meet the scalar multiplication condition, which states that d(kf(x))/dx = kd(f(x))/dxand ∫kf(x) dx = k ∫f(x) dx.
Points to Remember
- Integration by parts is a distinct mathematical approach for integrating two functions multiplied together.
- The technique to determine the derivative is to first simplify the provided statement by dividing it. The rule of indices is then applied to the resulting equation.
- Differentiation detects the rate of change of velocity with respect to time (known as acceleration).