Natural Logarithm and “e” = 2.71828……

e” has been an important constant that revolves around us all the time, especially during COVID-19 period where the cases are exponentially increasing at initial stages. Many mathematical memes i find in social media has differentiation of ex in it. I got the below meme from a social media friend.

Derivative of exponent meme

When i asked him what is “e” he said it is some constant, when i asked for its value he gave some GIF and EMOJI responses. But he knew the differentiation of ex is itself. End of my story.

This post and following post will help you to understand the value of “e” by tracing the path took by mathematicians to find it. And motivate you not to give GIF and EMOJI responses if someone asks you about this.

The content to follow will be divided into two parts:

Part 1 (Covered in this post):

  • Basics of differential calculus
  • Integration in terms of differentiation

Part 2 (Covered later):

  • Derivative (output after applying differentiation) of generic exponential function
  • Introduction to inverse function
  • Solving the famous “e” constant.

Differentiation basics

Differentiation gives the slope of any graph under study. Slope generally means change in “y” for change in “x”. If y = Distance and x = Time, then slope gives velocity. When algebra already gives us a formula for slope [(Y2-Y1) / (X2-X1)], it can help only when the graph is linear. Below graph shows velocity is constant at 1 m/s.

Linear graph
Figure [1]

But when graph is non linear, we can still use that same formula with “limit”ed modifications.

Dervitation concept
[2]
source: https://owlcation.com/

Slope of graph at a point = Slope of tangent at that point. When we try to find the slope of a tangent, calculus comes into picture. Calculus means “small stone” because it is to analyse by looking at elemental pieces. If you are not aware of secant and tangent, i would suggest you to quickly browse which wouldn’t take more than a minute. Assume a secant line is formed between (x,f(x)) and (∆x,f(∆x)).

Slope of secant line using algebraic formula= [ f(x+∆x) – f(x) ] / [(x+∆x) – x]

Secant becomes tangent when the change in y (∆y) and change in x (∆x) approaches 0. They approach 0 but don’t become 0. That’s why calculus is required (change in variable is very small). So formula for differentiation (denoted by dy/dx) is:

Derivation formula

Examples:

  • Differentiation of y=x (figure [1]) will be 1 as we already saw it.
  • Differentiation of y = x2 will be
Derivation of x squared
  • Differentiation of y = xn = nxn-1 (Proof would need knowledge on bernoulli’s formula and PASCAL triangle and much more, but i avoid here to be on track to destination “e” )
  • Differentiation of a constant will be 0 always since constant is 1st power of x.

Integration Basics

While differentiation gives slope of a graph, integration gives area under the graph. In figure [1] velocity remained constant at 1 m/s. From velocity and time, we could find the total distance as 1 m/s * 10s = 10 meters. This is same as area of velocity graph below = Length * breadth = 10 * 1 = 10 meters.

In previous section velocity was found by differentiating distance function. In this section distance is found by integrating the velocity function. So area of the slope function is same as the function itself. Read again if you didn’t get it 🙂 Hence integration is sometimes called as “anti-derivatives

Linear graph

Examples (compare with examples in differentiation section):

  • y = 1 (constant) for all x has integration output as x.
  • y = 2x has integration output as x2 (Area of triangle with base x and height 2x).
  • nxn-1 has integration output as xn ; So xn has integration output as (xn+1)/(n+1);
Note: I did not use calculus to define Integration, but i thought anti-derivation would be easy to grasp the concept.

So Part 1 is done. Before i close this post, i would like to conclude with an interesting motivation that led mathematicians to find the constant “e“. Integration of x to the power n gave us a neat formula. But here is a catch. If n= -1, what will be integration output? ∞? Think about it. Will see you in Part 2.

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