A body functions in respect to various other factors. Such factors influence the basic motion, stability and direction of that body. In order to understand the result of such factors operating upon a body, we use Calculus. To put it in simple terms, Calculus is that brunch of mathematics that deals with factors like limits, functions, derivatives, integrals and infinite series. These factors work based on several rules that governs the operation of calculus.
Factors of Calculus
- Differential Calculus: Differential calculus is concerned with the rate at which quantities change. The focus of the differential calculus is the derivative of a function. It is the rate of change of function near the input value.
- Derivative: Derivative measures how a quantity changes in response to change of different other factors. A moving object can change its direction based on friction and in respect of time. The net change of direction is the derivative. It is given as ∆y/∆x, where y=f(x). (where ∆y denotes change in y and ∆x denotes change in x)
- Integral: integration is just the reverse of differentiation. An integral ‘F(x)’ is achieved by integrating a differential f(x). The process is represented as ∫f(x) dx, where dx is the variable of integration.
Rules of Calculus
- The Product Rule: if we have to find the derivative of two or more functions, then we follow the product rule. It is represented as:
(f.g)’ = f’.g +f.g’
Or, d/dx (u.v) = u.d/dx (v) + v.d/dx (u)
- The Chain Rule: the chain rule finds out the derivative of a compound of two or more functions. It is given as:
derivative of h(x) = f(g(x)) in respect to ‘x’ is:
h’ (x) = f’ (g(x)) g’(x)
- The Inverse Function Rule: the inverse of a function is one such function that inverses the effect of the function. The inverse of function ‘f’ is given as ‘f-1’and it is represented as:
if a function ‘f’ has an inverse function ‘g’, given that g(f(x)) = x and f(g(x)) = y, then
g’ = 1/f’ . g
- The Polynomial or Elementary Power Rule: it is one of the most fundamental rules of calculus. It holds for a constant value of x0 and it is represented as:
d/dx (xn) = nx (n-1) where n not equal to 0
- The Quotient Rule: quotient rule is a method of finding derivative of function, which happens to be the quotient of two other functions for which the derivatives exists. It is represented as:
(f/g)’ = (f’ g- g’ f) / g2 where g is not equal to zero.
- The Reciprocal Rule: another short cut method for finding the derivative of a function that is a reciprocal of another differential function. It is given as:
h’ (x) = – [f ‘ (x)] / [f(x)]2 where h(x) = 1 / f(x)
Calculus is an important part of mathematical operations, since the times of Newton and Gottfried Leibniz during the 17th century. It is a well established branch of mathematics that finds prominence in application in elementary and higher education.