How would you first explain the meaning of derivative? A derivative is a formula that helps us to know about the changing relationship that occurs between two variable units. Let us consider the two variables a and b. The change in the value which is there in the dependent variable in regards to the change in the value of the independent variable expression can be estimated with the help of the derivative formula.

What Is the Use of the Derivative Formula?

Mathematically speaking, the derivative formula is very much useful in finding the slope of the line, finding the slope of the curved line, and also finding the change in one measurement in the change of other measurements. Next, let us learn about the derivative formula.

The Derivative Formula

The derivative formula is used as the recognition of a basic concept which is used in the study of calculus and in the process of finding a derivative, while this process is called the process of differentiation. This derivative formula is defined for the variable of x which has an exponent known as the ‘n’. This exponent can be an integer or a rational fraction. Thus the formula which helps in calculating the derivative is – ddx.xn = n.xn−1

What are the Other Formulae for Derivatives?

We will list down the different derivative formulae which are used in different fields of mathematics like in the study of calculus, trigonometry, and in other fields.  The formulae are as follows:

  • ddxddx.xn = n. xn-1
  • ddx.kddx.k = 0, where k is a constant
  • ddxddx.ex = ex
  • = ax. logee .a , where a > 0, a ≠ 1
  • ddxddx.logx = 1/x, x > 0
  • ddxddx. logaa e = 1/x logaa e
  • ddxddx.√x =1/(2 √x)

What are Differential Equations?

In the study of mathematics, there is an equation that contains one or more functions with the help of the derivatives. The derivatives which are present in a function define the rate of the change in the function. This is mainly used in the study of physics, engineering, biology, and other studies. The main purpose of the differential function is to study the solutions which properly satisfy the equations and all the properties of the solutions are also dealt with here.

This is one of the easiest ways to solve the differential equation which is explicitly used by using the formulae.

Definition of Differential Equation

A differential equation is an equation that consists of one or more than one term and the derivatives of one variable, this is the dependent variable, this is done with respect of some other variable this is the independent variable.

dy/dx = f(x) – In this equation x is the independent variable and y is the dependent variable here.

In a differential equation, there will either be partial derivatives of ordinary derivatives. The derivative represents the rate of change and the differential equation also describes the relationship between the quantity which is in constant change with respect to the change in any other quantity. There is also a lot of other differential equational formulae which help us in finding the solution of the derivatives. 

What Will be the Order of Differential Equation?

The order for the differential equation is the order which has the highest order derivative available in the particular equation. Here we will see some equations which represent the various orders. The equations are as follows:

  • dy/dx = 3x + 2 , The order of the equation is 1
  • (d2y/dx2)+ 2 (dy/dx)+y = 0. The order is 2
  • (dy/dt)+y = kt. The order is 1

This study was all based on the derivative formulae and differential equation. Students are required to understand the concept in-depth to solve the various equations and find out the dependent and independent variables. Likewise, other interesting mathematical concepts help us in gaining acute knowledge of the subject.

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