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## Difffferentiation Formulas, Integration Formulas

## Differential Calculus Formulas

Differentiation is a process of finding the derivative of a function. The derivative of a function is defined as y = f(x) of a variable x, which is the measure of the rate of change of a variable y changes with respect to the change of variable x. It means that the derivative of a function with respect to the variable x. There are a number of rules to find the derivative of a function. These rules make the differentiation process easier for different functions such as trigonometric functions, logarithmic functions, etc. Here, a list of differential calculus formulas is given below:

## Integral Calculus Formulas

The basic use of integration is to add the slices and make it into a whole thing. In other words, integration is the process of continuous addition and the variable “C” represents the constant of integration. But often, integration formulas are used to find the central points, areas and volumes for the most important things. Also, it helps to find the area under the curve of a function. There are certain important integral calculus formulas helps to get the solutions. These integral calculus formulas help to minimize the time taken to solve the problem. The list of integral calculus formulas is given below:

## 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) = lim
_{h→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(x^{n})/dx = nx^{n-1} | ∫x^{n} dx = x^{n+1}/(n + 1) + C, n ≠ -1 |

d(K)/dx = 0 | ∫K dx = Kx + C |

d(e^{x})/dx = e^{x} | ∫e^{x} dx = e^{x} + C |

d(a^{x})/dx = a^{x} log a | ∫a^{x} dx = a^{x}/log a + C |

d(ln x)/dx = 1/x | ∫(1/x) dx = ln x + C |

d(log_{a}x)/dx = 1/(x ln a) | ∫log_{a}x dx = x log_{a}x – 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^{-1}x, cos^{-1}x, tan^{-1}x, cot^{-1}x, sec^{-1}x, and cosec^{-1}x. 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 = sec^{2}x | ∫tan x dx = (1/a) ln |sec x| + C |

d(cot x)/dx = -cosec^{2}x | ∫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^{-1}x)/dx = 1/√(1 – x^{2}) | ∫sin^{-1}x dx = x sin^{-1}x + √(1 – x^{2}) + C |

d(cos^{-1}x)/dx = -1/√(1 – x^{2}) | ∫cos^{-1}x dx = x sin^{-1}x – √(1 – x^{2}) + C |

d(tan^{-1}x)/dx = 1/(1 + x^{2}) | ∫tan^{-1}x dx = x tan^{-1}x – (1/2) ln(1 + x^{2}) + C |

d(cot^{-1}x)/dx = -1/(1 + x^{2}) | ∫cot^{-1}x dx = x cot^{-1}x + (1/2) ln(1 + x^{2}) + C |

d(sec^{-1}x)/dx = 1/x√(x^{2} – 1) | ∫sec^{-1}x dx = x sec^{-1}x – ln(|x| + √(x^{2} – 1)) + C |

d(cosec^{-1}x)/dx = -1/x√(x^{2} – 1) | ∫cosec^{-1}x dx = x sec^{-1}x + ln(|x| + √(x^{2} – 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)

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.

## 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(x^{n})/dx = nx^{n-1}. Here n = -1. Therefore, we have

d(1/x)/dx = d(x^{-1})/dx

= -x^{-1-1}

= -x^{-2}

= -1/x^{2}

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 sec^{2}x.

**Solution:** To find the derivative of sec^{2}x, we will use the chain rule of differentiation. We have

d(sec^{2}x)/dx = 2 sec x × d(sec x)/dx

= 2 sec x × sec x tan x

= 2 sec^{2}x tan x

Next, for the integration of sec^{2}x, we know that differentiation and integration are reverse processes of each other and d(tan x)/dx = sec^{2}x. Therefore, we have

∫sec^{2}x dx = tan x + C, where C is the integration constant.

**Answer:** d(sec^{2}x)/dx = 2 sec^{2}x tan x and ∫sec^{2}x 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.

**1. What is Meant by Integration by Parts?**

Ans. Integration by parts is a unique mathematical technique of integration which is most commonly used when two functions are multiplied together but is also quite useful in other ways. It is actually a technique of using incorporating the product rule in reverse. Below is the rule of Integration by Parts:

Where, u is the function, denoted by u(x)

v is the function, denoted by v(x)

u’ is the derivative of the function, denoted by u(x)

Moreover,

The formula for Integration by Parts is as given:-

∫u dv = u. v −∫v du

**2. How do we Find the Derivative?**

Ans. You might find it really difficult, but it isn’t. The trick is to first simplify the given expression: perform the division (divide each term on the numerator by 3x^{½}. Use the law of indices in what equation we get. Now, differentiate term by term. Note that there are different ways of writing the derivative. They are although all typically the same. In other words:

(1) If y = x^{3},

dy/dx = 2x

This implies that if y = x^{3} the derivative of y, in respect to x is 3x

(2) d/dx (x^{3}) = 2x

This implies that the derivative of x^{3} in respect to x is 2x.

**What is Differentiation?**

Differentiation is the algebraic procedure of calculating the derivatives. The derivative of a function is the slope or the gradient of the given graph at any given point. The gradient of a curve at any given point is the value of the tangent drawn to that curve at the given point. For a non- linear curves, the gradient of the curve is varying at different points along the axis. Thus, it is difficult to calculate the gradient in such cases.

It is also defined as the change of a property with respect to a unit change of another property.

**What is Integration?**

Integration is the process to calculate definite or indefinite integrals. For some function f(x) and a closed interval [a, b] on the real line,

the definite integral,

is the area between the graph of the function, the horizontal axis, and the two vertical lines. These two lines will be at the endpoints of an interval.

When a specific interval is not given, then it is known as indefinite integral.

We will calculate the definite integral by using anti-derivatives. Therefore, integration is the reverse process of differentiation.

Remember that differentiation calculates the slope of a curve, while integration calculates the area under the curve, on the other hand, integration is the reverse process of it.

**Some Basic Differentiation Formula**

**Some Basic Integration Formula**

## Common Derivatives and Integrals

## Differentiation and integration rules from Calculus I and II

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