An introduction to Differential calculus.

Author- K.Venkataraman

Differential calculus is an area of math that focuses on rate of change and slope. In this blog, I will try my best to give the reader a brief understanding of the concept and give solved examples for better understanding. In this blog, I have taken excerpts from the first and second chapter of “Differential Calculus for Beginners” by Joseph Edwards and it is my wish that the reader understands the concepts discussed.

Limits.

Before we start about differentials, we will have to know the notion of a limit. The limit of a function for an assigned value of an independent variable is such that the value of the function may be to differ as less as possible by making the value of the variable approach the value of the function.

Eg.

Fig.1

Here in this limit, x is made, such that the value of x may differ as little as possible from 0 and not necessarily be equal to 0.

Solved examples

Fig.2

Important limits.

Here are some important limits to remember.

Fig.3

note- e= base of napierian logarithm.

Now I shall go to the concept of a differential.

Differentials.

a differential is the value of a slope in a graph, it is the tangent, as we know,

tanx= opposite side/hypotenuse

so, the following figure shall show the theorem.

Fig.4

So, the differential theorem can be used to show instantaneous rate of change

Fig.5

Here, Ф(x) is the function of x (ie.y) and Ф(x+h) is to show the addition of an infinitesimally small change (h).

Solved example: -

Question-(d/dx) *x^5

Fig.6

Quick math

Fig.7

Taylor expansions.

Trigonometric

expansions

There are many trigonometric functions in mathematics, for eg: sine, cosine, tangent. Etc. now, I shall explain the expansions of a couple functions.

Sine:

Fig.8

Cosine:

Fig.9

Tan^-1:

Fig.10

Now we shall see 2 algebraic expansions.

Algebraic expansions.

e^x

fig.11

loge(1+x)=

fig.12

now I hope the reader is clear with the concepts, because I will show some simple problems on the concepts and those interested can solve the problems.