Differentiation and Integration
Differentiation and integration are the important branches of calculus and the differentiation and integration formula are complementary to each other. On integrating the derivative of a function, we get back the original function as the result. In simple words, integration is the reverse process of differentiation, and hence an integral is also called the antiderivative. Differentiation is used to break down the function into parts, and integration is used to unite those parts to form the original function. Geometrically the differentiation and integration formula is used to find the slope of a curve, and the area under the curve respectively.
Further in this article, we will explore the differentiation and integration rules, formulas, and the difference between the two. We will also solve a few examples based on differentiation and integration for a better understanding of the concept.
What are Differentiation and Integration?
Differentiation and Integration are branches of calculus where we determine the derivative and integral of a function. Differentiation is the process of finding the ratio of a small change in one quantity with a small change in another which is dependent on the first quantity. On the other hand, the process of finding the area under a curve of a function is called integration. We can find the differentiation and integration of a function at particular values and within a particular range of finite limits. Integration of a function that is done within a defined and finite set of limits, then it is called definite integration.
The basic formula for the differentiation and integration of a function f(x) at a point x = a is given by,
- Differentiation: f'(a) = limh→0 [f(a+h) - f(h)]/h
- Integration: ∫f(x) dx = F(x) + C
Further, in the next section, we will explore the commonly used differentiation and integration formulas.
Differentiation and Integration Formulas
The differentiation of a function f(x) gives f'(x) which is the derivative of f(x), and further the integration of f'(x) gives back the original function f(x). Also sometimes the reverse process of integration is not able to generate the constant terms of the original function, and hence the constant 'C" is added to the results of the integration. In this section, we will explore the different most commonly used differentiation and integration formulas for algebraic functions, constant function exponential function, logarithmic function, and trigonometric functions.
| Differentiation | Integration |
|---|---|
| d(xn)/dx = nxn-1 | ∫xn dx = xn+1/(n + 1) + C, n ≠ -1 |
| d(K)/dx = 0 | ∫K dx = Kx + C |
| d(ex)/dx = ex | ∫ex dx = ex + C |
| d(ax)/dx = ax log a | ∫ax dx = ax/log a + C |
| d(ln x)/dx = 1/x | ∫(1/x) dx = ln x + C |
| d(logax)/dx = 1/(x ln a) | ∫logax dx = x logax - x/ln a |
Trigonometric and Inverse Trigonometric Functions Differentiation and Integration Formulas
Next, we will summarize all the trigonometric differentiation and integration formulas in the table below. We have six main trigonometric functions - sin x, cos x, tan x, cot x, sec x, and cosec x. Also, we will discover the formulas for the differentiation and integration of inverse trigonometric functions - sin-1x, cos-1x, tan-1x, cot-1x, sec-1x, and cosec-1x. The differentiation and integration of trigonometric functions are complementary to each other.
| Differentiation | Integration |
|---|---|
| d(sin x)/dx = cos x | ∫sin x dx = -cos x + C |
| d(cos x)/dx = -sin x | ∫cos x dx = sin x + C |
| d(tan x)/dx = sec2x | ∫tan x dx = (1/a) ln |sec x| + C |
| d(cot x)/dx = -cosec2x | ∫cot x dx = (1/a) ln |sin x| + C |
| d(sec x)/dx = sec x tan x | ∫sec x dx = (1/a) ln |sec x + tan x| + C |
| d(cosec x)/dx = -cosec x cot x | ∫cosec x dx = (1/a) ln |cosec x - cot x| + C |
| d(sin-1x)/dx = 1/√(1 - x2) | ∫sin-1x dx = x sin-1x + √(1 - x2) + C |
| d(cos-1x)/dx = -1/√(1 - x2) | ∫cos-1x dx = x sin-1x - √(1 - x2) + C |
| d(tan-1x)/dx = 1/(1 + x2) | ∫tan-1x dx = x tan-1x - (1/2) ln(1 + x2) + C |
| d(cot-1x)/dx = -1/(1 + x2) | ∫cot-1x dx = x cot-1x + (1/2) ln(1 + x2) + C |
| d(sec-1x)/dx = 1/x√(x2 - 1) | ∫sec-1x dx = x sec-1x - ln(|x| + √(x2 - 1)) + C |
| d(cosec-1x)/dx = -1/x√(x2 - 1) | ∫cosec-1x dx = x sec-1x + ln(|x| + √(x2 - 1)) + C |
Differentiation and Integration Rules
Further, we will go through some of the important and commonly used rules of differentiation and integration. The rules that are used for the differentiation of combinations of functions are product rule, quotient rule, and chain rule. Similarly, we use different rules for the integration of functions such as the fundamental theorem of calculus, and the commonly used methods of integration namely, substitution method, integration by parts, integration by partial fractions, etc. Given below are the rules of differentiation and integration and their formulas:
- Product Rule of Differentiation: [f(x)g(x)]' = f'(x)g(x) + g'(x)f(x)
- Quotient Rule of Differentiation: [f(x)/g(x)]' = [f'(x)g(x) - g'(x)f(x)]/[g(x)]2
- Chain Rule of Differentiation: [f(g(x))]' = f'(g(x)) × g'(x)
- The fundamental theorem of calculus 1 (FTC 1) is \(\dfrac d {dx}\) \(\int_a^x\) f(t) dt = f(x)
- The fundamental theorem of calculus 2 (FTC 2) is \(\int_a^b\) f(t) dt = F(b) - F(a), where F(x) = \(\int_a^b\) f(x) dx
- Integration by Parts: ∫f(x)g(x) dx = f(x) ∫g(x) dx - ∫[f'(x) × ∫g(x) dx] dx
Differentiation and Integration Difference
Now that we have understood the concept of differentiation and integration, let us now understand the differences between differentiation and integration. The table given below highlights their differences and the important properties of differentiation and integration.
Differentiation VS Integration
| Differentiation | Integration |
|---|---|
| Differentiation is a process of determining the rate of change in a quantity with respect to another quantity. | Integration is the process of bringing smaller components into a single unit that acts as one single component. |
| Differentiation is used to find the slope of a function at a point. | Integration is used to find the area under the curve of a function that is integrated. |
| Derivatives are considered at a point. | Definite integrals of functions are considered over an interval. |
| Differentiation of a function is unique. | Integration of a function may not be unique as the value of the integration constant C is arbitrary. |
Differentiation and Integration Similarities
Next, we will go through some of the common properties and similarities of differentiation and integration. The similarities and common formulas that are satisfied by both differentiation and integration are:
- They satisfy the property of linearity, 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 inverse processes of each other.
- They satisfy scalar multiplication property, that is, d(kf(x))/dx = kd(f(x))/dx and ∫kf(x) dx = k ∫f(x) dx
Important Notes on Differentiation and Integration:
- Differentiation and Integration are reverse processes of each other. Hence, on integrating the derivative of a function, we get back the original function as the result along with the constant of integration.
- Differentiation gives a small rate of change in a quantity. On the other hand, integration gives value over continuous limits and describes the cumulative effect of the function.
☛ Related Topics:
Differentiation and Integration Examples
-
Example 1: Evaluate the differential and integration of f(x) = 1/x.
Solution: To find the derivative of 1/x, we will use the differentiation formula d(xn)/dx = nxn-1. Here n = -1. Therefore, we have
d(1/x)/dx = d(x-1)/dx
= -x-1-1
= -x-2
= -1/x2
Next, for the integration of 1/x, we will use the formula ∫(1/x) dx = ln x + C. Therefore, we have
∫(1/x) dx = ln x + C
Answer: d(1/x)/dx and ∫(1/x) dx = ln x + C
-
Example 2: Determine the differentiation and integration of sec2x.
Solution: To find the derivative of sec2x, we will use the chain rule of differentiation. We have
d(sec2x)/dx = 2 sec x × d(sec x)/dx
= 2 sec x × sec x tan x
= 2 sec2x tan x
Next, for the integration of sec2x, we know that differentiation and integration are reverse processes of each other and d(tan x)/dx = sec2x. Therefore, we have
∫sec2x dx = tan x + C, where C is the integration constant.
Answer: d(sec2x)/dx = 2 sec2x tan x and ∫sec2x dx = tan x + C
FAQs on Differentiation and Integration
What are Differentiation and Integration in Calculus?
Differentiation and integration are the important branches of calculus and the differentiation and integration formula are complementary to each other. Differentiation is the process of finding the ratio of a small change in one quantity with a small change in another which is dependent on the first quantity. On the other hand, the process of finding the area under a curve of a function is called integration.
What is the Relationship Between Differentiation and Integration?
Differentiation and integration are inverse processes of each other. On integrating the derivative of a function, we get back the original function as the result, and hence an integral is also called the antiderivative. geometrically, differentiation gives the slope of a function whereas integration gives the area under the curve of the function.
How Are Differentiation and Integration Inverse Processes?
When we integrate the derivative of a function, we get the original function. Also, the integral of a function is called its antiderivative. The fundamental theorem of calculus gives the relationship between differentiation and integration, and shows how they are inverse processes of each other.
Why are Differentiation and Integration Linear Transformations?
Differentiation and Integration are Linear Transformations as they satisfy the following properties:
- 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
- d(kf(x))/dx = kd(f(x))/dx and ∫kf(x) dx = k ∫f(x) dx
What are the Differences and Similarities Between Differentiation and Integration?
Some of the common differences and similarities between differentiation and integration are:
- Differentiation is a process of determining the rate of change in a quantity with respect to another quantity and Integration is the process of bringing smaller components into a single unit that acts as one single component.
- They both satisfy the property of linearity.
- Derivatives are considered at a point and Definite integrals of functions are considered over an interval.
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