Differential Calculus is concerned with the rate of change of one quantity with respect to another quantity. A simple example would be acceleration. Acceleration is the rate of change of velocity with respect to the rate of change of time. 

For a function f(x), its differential equation would be:

f’(x) = dy/dx ……… (given that x≠0)

In the equation given above, 

LHS: f’(x) is the derivative of the function f(x)

RHS: y is a dependent variable and x is an independent variable

Limit is an important topic from calculus as it is further used to define concepts like continuity, integration, and differentiation. 

Consider the following function: f(x) = x2. Try putting various values in x and you will observe that, as the value of x gets closer to zero, the value of f(x) will simultaneously move towards 0. And hence, we say:

 x0f(x) = 0

We should read the above equation as: limit of f(x) as x tends to zero is equal to zero. 

Here, the left-hand side, that is, the limit of the function as x tends to 0, is considered as the value of the function should assume at x = 0. 

Example: If x2xn-2nx-2=80 and n is a positive integer, find n.

Solution: x2xn-2nx-2=80

∴n(2)n-1 = 80

∴ n(2)n-1 = 5(2)4

∴ n(2)n-1 = 5(2)5-1

∴n = 5

Let us now study and understand the difference between continuity and discontinuity.

When the left-hand limit of a function of f(x) as xmー and right-hand limit of a function f(x) as xm+ are equal, then the limit of f(x) at x=m exists. And if these limits are equal to f(m), then the function f(x) is said to be a continuous function at x=a.  

Discontinuity occurs for the functions which are defined only in a specific domain. An example of discontinuity would be f(x) = tanx. The tan function is defined in specific points and is discontinuous at these points. We can define discontinuity as follows:

If a function f(x) is not continuous at x = a, it is said to be discontinuous at x = a. And for the function f(x), ‘a’ is called the point of discontinuity.

Derivative is the fundamental concept of differential calculus. It gives the rate of change of functions. It provides the amount or rate at which the function changes at a given point. Derivative is nothing but a slope. It is used in measuring the steepness of the graph of the given function. Derivative of a function is the ratio of change of the value of a function with respect to the change in an independent variable. 

Example: If y = x-12×2-7x+5, then find dydx at x=2. 

Solution: y = x-12×2-7x+5= x-1(x-1)(2x-5)= 12x-5

dydx= (2x-5)d(1)d(x) -(1)d(2x-5)dx(2x-5)2= (2x-5)(0) -(1)(2)(2x-5)2= -2(2x-5)2

For dydx= -2(2x-5)2at x=2,

dydx= -2(4-5)2 = -21= -2

∴dydx= -2

Integral calculus is nothing but the process of finding the area under a curve. In integral calculus, to find the area under the curve exactly, we divide it into infinite rectangles that have infinitesimal width and sum them up. In integral calculus, we will study the integrals and their different properties.

Integration is nothing but uniting or bringing the coordinates of a function to sum them up. It is the reverse process of differentiation. In integration, we add discrete and small data, which cannot be added particularly, and then represent it in a single value. A very simple daily life example would be a loaf of bread. When the loaf is cut into multiple slices, we differentiate them. When these slices are put together, we integrate them. 

Example: Find (tanx-cotx)2dx

Solution:

(tanx-cotx)2dx = (tan2x-2tanxcotx+cot2x)dx

= (sec2x+cosec2x-4)dx

= sec2x.dx + cosec2x.dx -41.dx

= tanx-cotx-4x+c

∴(tanx-cotx)2dx = tanx-cotx-4x+c

1. What is differential calculus?

A: The calculation of the rate of one quantity with respect to another quantity is called differential calculus.

2. What is a derivative?

A: Derivative (dydx) is nothing but the slope. It gives the measure of the steepness of the graph of a function. 

3. What are the real-life applications of differential calculus? 

A: For calculating the rate of change of temperature, for calculation of distance or speed covered. 

4. What is a differential equation? 

A: Functions and their derivatives that are related to each other in the form of an equation is a differential equation. 

5. What is integration?

A: Integration is nothing but uniting or bringing the coordinates of a function to sum them up. It is the reverse process of differentiation.