In pr*e*vious post i conclud*e*d with an op*e*n qu*e*stion of “int*e*gration of x^{-1}” and if you w*e*r*e* awar*e* of calculus formula you would have guessed right. This post r*e*volv*e*s around a path to find th*e* solution, l*e*ading to famous “*e*“.

Whil*e* one part of math*e*maticians w*e*r*e* puzzl*e*d about this qu*e*stion, th*e*r*e* w*e*r*e* oth*e*r s*e*t of math*e*maticians working on diff*e*r*e*ntiation of functions known to th*e*m. Th*e*y w*e*r*e* not worri*e*d much about this int*e*gration of x^{-1}, th*e*y k*e*pt inv*e*nting n*e*w formulas for diff*e*r*e*nt functions. On*e* such function was Expon*e*ntial function.

Diff*e*r*e*ntiation formula can b*e* us*e*d to calculate the slop*e* of any *e*xponential function of form y = b^{x} . Unlik*e* pow*e*r function which has variabl*e* in bas*e* and constant in pow*e*r (Eg: y=x^{2}), *e*xpon*e*ntial function has constant in bas*e* and variabl*e* in pow*e*r. High*e*r th*e* valu*e* of bas*e*, fast*e*r th*e* incr*e*as*e* in “y”.

Th*e*r*e* ar*e* two IMPORTANT inf*e*r*e*nc*e*s mad*e* from abov*e* equations, which will b*e* r*e*f*e*rr*e*d lat*e*r in this post (r*e*ad twic*e* if it n*e*c*e*ssitat*e*s):

**Slop**.*e*of b^{x}is proportional to its*e*lf- “c” is a constant which is
**Slop**and its valu*e*of b^{x}at x=0*e*d*e*p*e*nds on base “b”.

Park in mind thos*e* two inf*e*r*e*nc*e*s until inv*e*rs*e* function is bri*e*f*e*d. Math*e*maticians w*e*r*e* awar*e* of inv*e*rs*e* functions and th*e*ir r*e*lation with actual function using diff*e*r*e*ntiation. Inv*e*rs*e* functions ar*e* thos*e*, which will r*e*v*e*rs*e* th*e* output back to its input. Eg: f(x) = y = 2x, inv*e*rs*e* function will b*e* g(y) = y/2 which will giv*e* back x. R*e*lationship b*e*tw*ee*n d*e*rivative of a function and d*e*rivativ*e* of its inv*e*rs*e* function can b*e* found using chain rul*e* of diff*e*r*e*ntiation. In pr*e*vious *e*xampl*e* dy/dx = 2; dx/dy = 1/2; Tak*e* any function and its inv*e*rs*e*, th*e*ir d*e*rivativ*e*s multiply to produc*e* id*e*ntity. In oth*e*r words, **slop e of an inverse**

**function is “-1 pow**

*e*r of slop*e*of its normal function”.Now going back to our *e*xponential function. We diff*e*r*e*ntiated b^{x} and *e*nd*e*d up in “c” which is the slop*e* of function at x=0. Inv*e*rs*e* function for b^{x} is log_{b}(y). If you ar*e* surpris*e*d on this sudd*e*n magic, substitute b^{x} into log_{b}(y) => log_{b}(b)^{x} = xlog_{b}(b) = x;

Note: log_{b}(b) = 1; log_{b}(1) = 0;

Logarithm is anoth*e*r int*e*r*e*sting topic which would n*ee*d anoth*e*r s*e*parat*e* post to cov*e*r. For now l*e*ts not d*e*viat*e* from our obj*e*ctiv*e* but r*e*m*e*mb*e*r the bas*e* of the logarithm “b”.

In *e*xpon*e*ntial world x = 0 giv*e*s y = b^{0} = 1; In logarithm to th*e* bas*e* “b” world y = 1 gives x = 0; Valu*e* of x and y r*e*mains sam*e* in both worlds but the d*e*p*e*nd*e*nt variabl*e* will b*e* x in first cas*e* and y in its inv*e*rs*e* cas*e*.

“*c*” in *e*xpon*e*ntial world is slop*e* of curv*e* when x = 0; “1/c” will b*e* the slop*e* at y = 1 in logarithm to bas*e* “b” world; How?

W*e* already s*ee*n the r*e*lationship b*e*tw*ee*n d*e*rivativ*e*s of normal and its inv*e*rs*e* function, which will h*e*lp to find th*e* derivativ*e* of log_{b}(y) *e*asily without much computation.

So slop*e* of the *e*xpon*e*ntial function at x = 0 is “-1 power of” slop*e* of its inv*e*rs*e* function at y=1;

Ph*e*w ! Now, do you r*e*m*e*mb*e*r the qu*e*stion ask*e*d in part 1? What is anti-d*e*rivativ*e* of x^{-1}? W*e* ar*e* now just on*e* st*e*p ah*e*ad of m*ee*ting our boss “*e*“.

Log_{b}(y) gav*e* d*e*rivation as 1/cy. Log_{b}(x) will giv*e* d*e*rivation as 1/cx (Just a chang*e* in variabl*e* do*e*s not cost anything). In som*e* way if w*e* make c=1, th*e*n anti-d*e*rivativ*e* of 1/x can b*e* found.

We hav*e* two final tasks now:

- D
*e*riv*e*formula for “c”. - Sinc
*e*“c” d*e*p*e*nds on bas*e*“b”, comput*e*“b” to mak*e*“c” = 1.

The valu*e* insid*e* par*e*nth*e*sis should b*e* constant sinc*e* it involv*e*s only ∆x which t*e*nds to 0. Tim*e* to r*e*v*e*al the formula for “*e*“.

“*e*” was chos*e*n by Swiss math*e*matician “*E*ul*e*r” and popularly call*e*d as *euler constant.*

First inference : "e" is a mathematical constant.

Second inference: "e" is also the baseof the logarithm in order to make"c" = 1.

Logarithm to th*e* bas*e* “*e*” is known as Natural logarithm and most t*e*xtbooks r*e*f*e*r it as ln(x). This rais*e*s curtain for the qu*e*stion ask*e*d b*e*for*e*. Anti-d*e*rivativ*e* of x^{-1} is ln(x).

Third inference: In ourexponential function b^{x}, if b =e, then thederivativeoutput will beitself

R*e*ason for th*e* t*e*rm “natural” is, *e*xpon*e*ntial growth app*e*ars in most of natural occurring ph*e*nom*e*non around us. Population growth in any y*e*ar d*e*p*e*nds on the total population in that y*e*ar. A hot cup of coff*ee* will cool down at a rat*e* proportional to the temp*e*rature at that point.

Aft*e*r a small and quick chang*e* of variabl*e*s, *e* can b*e* updat*e*d as b*e*low:

n | 1/n | 1 + 1/n | (1 + 1/n)^{n} |
---|---|---|---|

1 | 1 | 2 | 2 |

2 | 0.5 | 1.5 | 2.25 |

10 | 0.1 | 1.1 | 2.593742 |

100 | 0.01 | 1.01 | 2.704814 |

1000 | 0.001 | 1.001 | 2.716924 |

10000 | 0.0001 | 1.0001 | 2.718146 |

100000 | 0.00001 | 1.00001 | 2.718268 |

1000000 | 0.000001 | 1.000001 | 2.718280 |

10000000 | 0.0000001 | 1.0000001 | 2.718281 |

Fourth inference: As "n" tends to higher value,etends to its actual value. With abovevalueof n= 10 million,eis 2.71828....

Finally, the job is don*e*. B*e*low mind map summariz*e*s the path tak*e*n to r*e*ach “*e*“.

B*e*fore i clos*e* this post, i would lik*e* to put the g*e*n*e*ral formula for diff*e*r*e*ntiation of *e*xpon*e*ntial function.

For the sake of brevity, we will always represent this number 2.71828… by the letter

– Leonhard Eulere. Now i will have less distraction.

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