Mastering Differential Calculus: A Comprehensive Guide

What is differential calculus? For the differential calculus  basics, you will have to go through differential calculations, derivatives, and their applications. We understand differentiation as a process of finding the derivative of a function, and we understand the derivative of a value as the rate of change of a function in terms of the given values for any value. 

Let us go through some basic concepts of differential calculus to figure out what is differential calculus and calculus functions

Function

A function is a binary relation where each input is mapped to exactly one output.

 For example, let y=8x-3 be a function. 

Here, 

For any value of ‘x,’ we get a single value for ‘y.’

Dependent And Independent Variable

A dependent variable is a variable whose value is determined by another variable known as the dependent variable. It is also called the Outcome Variable, where the outcome is evaluated from a mathematical expression of the independent variable. 

The independent variable can be defined as a variable whose value does not depend on any variable. 

For example;

In the function y=8x-3, ‘y’ is a dependent variable as the value of ‘y’ depends on what the value of ‘x’ will be, and ‘x’ is the independent variable.

If the value of ‘x’ is 2, the value of ‘y’ will be 13. 

Domain and Range

We can define the domain of a function as a set of values that we put into the function, and the range is the value that we get once we input the value in the function. 

Let’s take an example f(x)=5x 

For a set of values {1,2,3}, which will be the domain, the range will be

f(1)= 5*1= 5

f(2)= 5*2= 10

f(3)= 5*3= 15

Hence, the range will be a set of values {5,10,15} for the function f(x)=5x.

Limits

Limits are an important part of calculus. They define continuity, intervals, and derivatives in calculus. We define the limit of a function in the following manner: 

Let us take a function “f”, whose limit is “z”, which is a real number. Then, we define the limit as 

limx→cf(x)= L 

It is expressed as “the limit of a function of x, as x approaches z is equal to L”. The “lim” represents the limit, and the arrow represents the fact that x approaches z. 

Intervals

Intervals are defined as a set of real numbers that lie between any two specific numbers of the set R (Real Numbers). 

Let’s assume two real numbers, ‘a’ and ‘b’, where a<b. Then, we use different interval notations so that we can define different types of intervals called notations. 

Interval Notations is a method of representing the subset of real numbers bounded between two numbers. 

For example; 

5<x<10 is an interval. Now we see that ‘x’ lies between 5 and 10 but will not be equal to 5 or 10. 

There are different types of Interval Notations. They have been defined below. 

1. Open Interval: The set of real numbers {x: a < x < b} is defined as an open interval. We represent open intervals by (a,b). An open interval contains all points between ‘a’ and ‘b’, excluding ‘a’ and ‘b’ themselves. 
   
   
2. Closed Interval: The set of real numbers  {x: a ≤ x ≤ b} is defined as a closed interval. It is represented by [a,b]. The closed intervals contain all points between ‘a’ and ‘b’, including ‘a’ and ‘b’. 

Derivatives 

Derivatives are a fundamental tool of differential calculus. It shows us the momentary change a function undergoes at any point in time. It can be defined as the ratio of change in the value of a function to the independent variable, i.e., dy/dx. The derivative is a slope measuring the steepness of the graph of the function. 

Definition of a derivative with the help of a graph: It is the point where the slope of the tangent coincides with the point on the curve.