Calculus was initially known as infinitesimal calculus or “the calculus of infinitesimals”. Infinitesimals calculus came about in the 17th century. It was developed by Isaac Newton and Gottfried Wilhelm Leibniz.

Calculus is a Latin word meaning “small stones.” It is called so because it is like using small pebbles for calculating something. Differentiation in calculus cuts something into small bits to know about its changes. Integration in Calculus joins the small bits together to know the quantities.

Calculus is the study of continuous change.

The two major branches used in calculus are Differentiation and Integration. However, it is difficult to understand the difference between differentiation and integration. Many students and even scholars are not able to understand its difference.

**Differentiation vs Integration**

The difference between Differentiation and Integration is that differentiation is used to find out the instant rates of change and the slopes of curves, whereas if you need to calculate the area under curves then make use of Integration. As you can see, both differentiation and integration are opposite to each other in mathematical significance.

## Comparison Table Between Differentiation and Integration

Parameters of Comparison | Differentiation | Integration |
---|---|---|

Purpose | Differentiation is used to calculate the gradient of a curve. It is used to find out the instant rates of change from one point to another. | Integration is used to calculate the area under or between the curves. |

Real-life application | Differentiation is used to calculate instant velocity. It is also used to find whether a function is increasing or decreasing. | Integration is used to calculate the area of curved surfaces. It is also used to calculate the volume of objects. |

Addition and division | Differentiation uses division to calculate the instant velocity or any desired results. | Integration uses addition for its calculations. |

Directly opposite | Differentiation is the reversed process of integration. | Integration is the reversed process of differentiation. |

Role | Differentiation is used to calculate the speed of the function as it calculates instant velocity. | Integration is used to calculate the distance covered by any function as it calculates the area under the curve. |

## What is Differentiation?

In mathematics, the method of finding the rate of change of a function or finding the derivative is known as Differentiation.

The three derivatives are:

- Algebraic functions-
*D*(*x*) =^{n}*nx*^{n}^{ − 1 } - Trigonometric functions-
*D*(sin*x*) = cos*x* - Exponential functions-
*D*(*e*) =^{x}*e*^{x}

Differentiation is used to calculate the gradient of a curve and to find out the instant rates of change from one point to another.

There is a ‘chain rule’ which helps to differentiate composite functions. Calculation of instant velocity is one of the real time use of differentiation.

## What is Integration?

In calculus, the integration refers to the formula and the method used to calculate the area under the curve. It is used to calculate so because it is not a perfect shape for which the area can simply be calculated. Just like differentiation, integration also has real-life applications. It is used to calculate the areas of curved surfaces. It helps in calculating the volume of objects.

Integration is used to find the distance moved by any function. The distance traveled by the function is the area under the curve. This area is calculated using the algebraic expression Integration. It obtains the desired result using addition.

**Main Differences Between ****Differentiation and Integration**

**Differentiation and Integration**- Integration and differentiation are mainly different in the way both are applied and their final results. The main difference is that they are used to obtain different answers. It is difficult to obtain the gradients of non-linear curves as they have different slopes at any given point. Here differentiation comes in. It is used to calculate the gradient of the curve. Differentiation is the algebraic expression that is used to know the change incurred from one point to another.
- On the other hand, the algebraic expression used to calculate the area under or between the curves is Integration.
- Both Differentiation and Integration are important calculus concepts that are used in real-life scenarios.
- Differentiation is mainly used for calculating instant velocity. It is used to know whether the function is increasing or decreasing.
- Integration is used to calculate the area of curved surfaces. It is used to calculate the volume of different objects.
- Another point of difference between differentiation and integration is the method they use for calculations. The results of differentiation are obtained through division, whereas the results of integration are obtained by addition.
- Both Differentiation and integration are the direct opposite of one another. If one is using differentiation, he is said to be using the opposite of integration. Similarly, if one is using integration, he is said to be using the opposite of differentiation.
- Differentiation is used to calculate the speed of the function as it calculates instant velocity, whereas integration is used to calculate the distance covered by any function as it calculates the area under the curve.

## Conclusion

One of the main differences between differentiation and integration is that the two algebraic applications are the direct opposite of each other in their application.

It is very important to understand the concept and difference of both of them in order to obtain the results of the functions and in order to know where to apply which algebraic expressions.

It is also important to understand the two calculus concepts as they are broadly used in various disciplines like business applications, economics applications, and engineering.

Basically, differentiation is used to calculate the gradient of a curve and it is used to find out the instant rates of change from one point to another whereas Integration is used to calculate the area under or between the curves.

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