The process by which the derivatives of a function are determined can be defined as differentiation and the formula used is known as the differentiation formula. It is a concept that is present in the calculus of mathematics. Now, the derivative is the rate at which a function changes concerning the time given. For example, velocity is the rate of change of displacement concerning time. Similarly, speed is the rate of change in distance with respect to time. However, the speed at each instance is not equivalent to the other as the average is calculated.
Therefore, we can say that the speed is considered the same as the slope, that is nothing, but the instant change of distance over a period. The laws of differentiation and derivatives were given by Sir Isaac Newton.
The concepts and principles of derivatives are used in many disciplines of science. The opposite of differentiation is integration. Both of these concepts are widely used in the field of mathematics. They are considered the major concepts of calculus. In this article, we shall cover some interesting topics.
Some of them are: what are integration, methods of integration, and application of derivatives.
The process of finding the area of the surface under the curve can be defined as integration. We can also define integration as the process of uniting the part of a whole region or surface. While we solve the integration in calculus, the differential of the function is given. This signifies that integration is the inverse of differentiation of vice-versa. Integration is mostly used to find the area under the region of a curve that is bounded by the functions.
Now, the area of the curved shape is determined by knowing the total number of polygon sides that are inscribed in the curve. This whole process is defined as the method exhaustion, later on, it was given the name, integration. We have already discussed the differentiation in the above section. Hence, we can say that differentiation and integration are the sole of calculus and are used to solve problems based on physics and mathematics. Leibniz was the scholar who proposed and came up with the principles of integration.
Methods of Integration
As mentioned above, the process of finding the area of the surface under the curve can be defined as integration. Most of the time, an inspection of the function is not enough to determine the integral of some function. Hence, we use some additional methods to determine or find the integrals of functions. Some of those additional methods are the decomposition method, integral by parts, substitution method, and so on. The following points are the methods of integration in detail.
- The method where the function can be decomposed into the difference or sum of the function can be defined as the integration by decomposition. In this process, the individual integrals are determined or known at first. The integrand can be algebraic, exponential, and trigonometric or can be a mixture of these functions.
- A method of integration where we can change the variable of integration in an order such that the integrand is integrated easily and effectively. This process is known as integration by substitution. You must note that these functions can use trigonometric identities for substitution.
Application of Derivatives
As mentioned above, the derivative is the rate at which a function changes concerning the time given. The following points signify the application of derivatives.
- The foremost use of derivatives is to determine or find the rate of change of a quantity concerning the other quantity.
- Derivatives are also used to find the equation of a tangent and the normal of a curve.
- The linear approximation of a given function concerning a given value can be calculated via derivatives.
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