Differential Calculus

The subfield of mathematics which is mainly deals with study of the rates at which the change of quantities takes place is referred to as differential calculus. There are two traditional divisions of calculus namely integral calculus and differential calculus. The rate of change of function at a par derivative of a function

Theorem of calculus:

The fundamental theorem of calculus connects the integral calculus and the differential calculus. According to this theorem the reverse process to the integration is referred to as differentiation. The application of the differentiation is present in almost all quantitative discipline. The displacement of a moving body’s derivate with respect to time is referred to the velocity of the body. On the other hand the acceleration is the derivative of the acceleration in accordance to the time. According to the Newton’s second law of motion, the derivative of momentum of the body is equivalent to that of the force that is applied to the body. In the field of operation research, the most effective ways of the design factories and the transport materials are determined by the derivatives.

The minima and the maxima of a function are often found using these derivatives. The equation that involves the derivatives is referred to as differential equations and these equations are the fundamental thing in describing the natural phenomena. 

In various fields of mathematics like the functional analysis, complex analysis, measure theory, differential geometry and abstract algebra these derivatives and the generalization of these derivatives are found.

If x and y are real numbers where y is a function of x and so for every x value there would certainly be a corresponding y value. There by the relationship is written in the form y = f(x).  For instance, if f(x) be the equation for the straight line, then two real numbers namely m and b will be y = m x + b. In here, m is the slope which can be identified with the help of the formula.

m = (change in y / change in x) = Δy / Δx ,

The symbol Δ is an abbreviation for "change in". There by it follows that Δy = m Δx.

Differential equations:

The relation that exists between the collection of the functions and their derivatives is referred to as the differential equation of differential calculus. There are two differential equations namely ordinary differential equation and the partial differential equation. The ordinary differential equation relates the functions of one variable to their relative in accordance with that variable where as the partial differential equation relates functions of more than one variable.

Questions:

• What is differential calculus?
• What is differential equation?