## Differentiation and Integration

Calculus, originally called infinitesimal calculus or “the calculus of infinitesimals”, is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Two mathematicians, Isaac **Newton of England** and **Gottfried Wilhelm Leibniz** of Germany, share credit for having independently developed the calculus in the 17th century.

Calculus is one of the most important branches of mathematics that deals with continuous change. The two major concepts that calculus is based on are derivatives and integrals. The derivative of a function is the measure of the rate of change of a function, while integral is the measure of the area under the curve of the function. The derivative gives the explanation of the function at a specific point whereas the integral accumulates the discrete values of a function over a range of values.

Calculus in Mathematics is generally used in mathematical models to obtain optimal solutions and thus helps in understanding the changes between the values related by a function. Before calculus was invented, all math was static: It could only help calculate objects that were perfectly still. Calculus is broadly classified into two different sections:

- Differential Calculus (Differentiation)
- Integral Calculus (Integration)

Differentiation and integration are inverses to each other. While differentiation helps you find derivatives of functions and examine, if the functions are continuous and differentiable at particular points, if they have a saddle point, integration helps you find the summation of a function over a defined region and helps you find the area covered below the curve, common area to the two curves, if the curve exists within certain limts, or the area occupied in that definite limit.

Calculus is used in a multitude of fields that you wouldn’t ordinarily think would make use of its concepts. Among them are physics, engineering, economics, statistics, and medicine. It makes it possible to solve problems as diverse as tracking the position of a space shuttle or predicting the pressure building up behind a dam as the water rises. Computers have become a valuable tool for solving calculus problems that were once considered impossibly difficult.

### Differentiation and Integration Formulas

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(x^{n}) = nx^{n-1} | ∫ x^{n} dx = (x^{n+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 = sec^{2} x | ∫ sec^{2} x dx = tan x + C |

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

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

## Differentiation vs Integration: Key Differences

Points 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. |

## Key Takeaways

- Calculus is the study of rates of change.
- There are two types of calculus: Differential calculus determines the rate of change of a quantity, while integral calculus finds the quantity where the rate of change is known.
- Gottfried Leibniz and Isaac Newton, 17th-century mathematicians, both invented calculus independently. Newton invented it first, but Leibniz created the notations that mathematicians use today.
- Differentiation deals with the calculation of a derivative which is the instantaneous rate of change of function taking into one of its variables into consideration.
- Integration is able to determine the function provided its derivative. It measures the area under the function between limits.
- Differential calculus determines the rate of change of a quantity. It examines the rates of change of slopes and curves.
- Integral calculus focuses on such concepts as slopes of tangent lines and velocities. While differential calculus focuses on the curve itself.
- Integral calculus concerns itself with the space or area
*under*the curve. Integral calculus is used to figure the total size or value, such as lengths, areas, and volumes.