Calculus was initially known as infinitesimal calculus or “the calculus of infinitesimals”. Infinitesimals calculus came about in the 17th century.

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. Many students and even scholars are not able to understand its difference.

## Key Takeaways

- Differentiation is a mathematical operation in calculus that calculates a function’s rate of change or slope at a specific point.
- Integration is the inverse operation of differentiation, computing the accumulated sum of a function’s values over a given interval, used to find areas, volumes, or other quantities.
- Both differentiation and integration are essential concepts in calculus, but they serve opposite purposes, with differentiation focusing on rates of change and integration on accumulation.

**Differentiation vs Integration**

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

## Comparison Table

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. | The 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, 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 site can be calculated.

Integration is used to find the distance moved by any function. The distance travelled by the function is the area under the curve.