The comparison of quantities which varies in a one-liner way is known as Calculus. Calculus is known to have significant applications in Science and Engineering. Quite a few topics such as acceleration of current in a circuit or velocity are known not to behave in a linear fashion. Therefore, we need calculus to study the continuously changing quantities. We can also say that calculus is that branch of mathematics which deals with continuous change. In this article, we will primarily look into a brief introduction to Differential Calculus. Let us first understand, what is it.
There are two main branches of Calculus
- Differential Calculus- Differential Calculus concerns itself with the study of the rates at which quantities change.
- Integral Calculus-Integral Calculus concerns itself with the accumulation of quantities and areas within the curves.
The ancient mathematician-Aryabhatta had used differential calculus to find the motion of the moon. Contemporary mathematics uses differential equations and functions to find out the maxima and minima of curves which are in common usage.
Some more applications are as follows
- Water mixers
- Automatic air conditioners- temperature control
- Cruise control in cars
- Industrial control systems rockets, ships etc
The degree of closeness to any value or approaching term is known as limits. Limits are an important part of calculus and mathematical analysis and used to define integrals, derivatives, and continuity.
The limits are expressed as
It is read as “the limit of f of x as x approaches c equals A”. The “lim” shows limit, and fact that function f(x) approaches the limit A as x approaches c is described by the right arrow as
The study of the definition, properties, and applications of the derivative of a function is known as differential calculus. The process of finding the derivative is called differentiation.
The expression for the derivative is defined as-
Derivative in Differential Calculus
‘y’ is a dependent variable and ‘x’ is an independent variable.
If there is a change in the value of x that is the change in x will bring a change in y, let that be.
Now to find out the change in y with a unit change in x can be found as follows:
It is called the derivative.
Let f(x) be a function where the value varies since the value of x varies
Steps to find the Derivative:
First you need to change x by the smallest possible value and let that be (dx) and so the function becomes f(x+dx)
Thereafter, get the change in value of function that is : f(x + dx) – f(x)
You will see that the rate of change in function f(x) on changing from ‘x’ to ‘x+dx’ will be
f(x + dx) – f(x)
Now d(x) is ignorable because it is considered to be too small.
Example: Let f(x) be a function where f(x) = x2
The derivative of x2 is 2x means that with every unit change in x, the value of the function becomes twice (2x).
The process of finding the area under the curve is known as Integration. You can use integration to find out areas, volumes etc.
Integration can be of two types
- Definite Integral- where the limits are defined.
- Indefinite Integral- where limits are not defined.
Differentiation is defined as the derivative of a function regarding the independent variable and it can be applied to measure the function per unit change in the independent variable.
If y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by dy / dx
If the function f(x) undergoes a minute change of h near to any point x, then the derivative of the function is defined as
in case a function is denoted as y=f(x), then the derivative is indicated by the following notations.
D(y) or D[f(x)] is called Euler’s notation.
dy/dx is called Leibniz’s notation.
F’(x) is called as Lagrange’s notation.
Differentiation is the process of determining the derivative of a function at any point.
Functions in calculus can be categorized under two heads
(i) Linear function
(ii) Non-linear functions
Linear function is that which varies with a constant rate through the domain. The overall rate of change of function remains the same at any given point. The rate of change though may vary from point to point in those cases of non-linear functions.
The essential parts of a differential equation are functions and their derivatives. The function represents the physical quantities. The derivative represents the rate of change and their relationship is represented by the differential equation.
Isaac Newton gave three kinds of differential equation-
Some of the important Differentiation formulas in differentiation are as follows.
If f(x) = tan (x), then f'(x) = sec2x
If f(x) = cos (x), then f'(x) = -sin x
If f(x) = sin (x), then f'(x) = cos x
If f(x) = ln(x), then f'(x) = 1/x
If f(x) = ex, then f'(x) = ex
If f(x) = xn, where n is any fraction or integer, then f'(x) = nxn−1
If f(x) = k, where k is a constant, then f'(x) = 0
Some of the basic differentiation rules that need to be followed are as follows.
(i) Sum or Difference Rule
If the function is sum or difference of two functions, the derivative of the functions is the sum or difference of the individual functions, i.e.,
(ii) Product Rule
If the function f(x) is product of two functions u(x) and v(x), the derivative of the function is,
(iii) Quotient rule
If the function f(x) is in the form of two functions [u(x)]/[v(x)], the derivative of the function is
(iv) Chain Rule
If a function y = f(x) = g(u) and if u = h(x), then,
This was a consolidated study of the basics of Differential Calculus. If you want to know how Calculus is applicable in our daily lives, visit here.