The differential equation is an important concept in mathematics, precisely calculus. It is applied in various fields like physics, biology, engineering, etc. As per definition in mathematics, a differential equation is an equation that contains one or more functions with their derivatives. The derivatives of a function refer to the rate of change of that function at a certain point. The purpose of a differential equation is to find out the conditions or solutions that satisfy this equation.
A differential equation describes the relationship between a quantity that is continuously changing and the rate of change of another quantity. In calculus, the differential equation is represented by the derivatives of one variable (known as the dependent variable) for the change in another variable (known as the independent variable). The expression of a differential equation is f(x) = dy/dx where y is the dependent variable and x is the independent variable. The rate of change of x is denoted by dx and the rate of change in y is denoted by dy.
The two most common types of differential equations are ordinary differential equation and partial differential equation which are explained as follows:
- Ordinary differential equation: The function involved in this equation depends on a single variable and its derivatives are called ordinary derivatives.
- Partial differential equation: The function involved in this equation depends on several independent variables and its derivatives are partial derivatives.
The order of a differential equation is determined by the highest degree or power of the derivative present in the equation. The differential equation having the order of 1 is called the first-order differential equation. It is to be noted that the order of the differential equation will always be positive integers.
The solutions of the differential equation give another function that can be used to find the property of the original function. The solution of the differential equation is a function of the dependent variable in terms of the independent variable. This solution satisfies the differential equation. The process of finding a solution to a differential equation is called integration.
The derivative formula refers to a set of rules that helps to find out solutions for problems based on differential equations. The formula includes a list of derivatives for constants, polynomials, trigonometric functions, exponentials, hyperbolic functions, and logarithmic functions. The derivative formula is an important concept used in calculus.
Derivatives describe the changing relationship between two variables one of which is the dependent variable and the other is the independent variable. The derivative formula is used to find the derivatives which indicate a change in values of the dependent variable for the change in the independent variable. The process of finding derivatives with help of the derivative formula is known as differentiation. In Mathematics, the derivative formula defines the amount by which a function changes at a given point in time. The derivatives are often represented as dy/dx of f(x).
The concepts of derivatives and derivative formula are used to solve different problems related to real-life problems such as calculating the variation in temperature, determining the speed or distance covered, profit or loss, etc. Some of the problems involve deduction of derivatives of complex functions which becomes easier with help of a host of derivative formulas and properties that we can use to simplify the problems considerably.
- The constant rule states that the derivative of a constant is zero. It is expressed as d/dx(K)=0
- The power rule states that the derivative of a variable x with exponent n will be n multiplied by x raised to the power of (n-1).
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