# Derivatives and exponential function

• icystrike
In summary, the conversation discusses the concept of M(e) and its relation to the derivative of the exponential function, e^x. It is determined that M(e) is equal to 1 and discussion ensues about why this specific value was chosen. The explanation is that this value of e is the one for which the area under the function f(x)=1/x between x=1 and x=e is equal to 1.

## Homework Statement

I don't understand why they took M(e)=1 , and how the proceed on with the proof.

## The Attempt at a Solution

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Try to write the derivative of the exponential with the limit definition:

f'(x)=lim h->0 (f(x+h)-f(x))/h

and try to factor out e^x

yes. i understand . However, why did they take M(e)=1?
I can't really accept their explanation : M(2)<1 and M(4)>1 ,thus they allow M(e)=1..
My question is how about other real number between 2 to 4?

Last edited:

It's NOT so much "how" as "why". I presume that just before the section you quoted they showed that the derivative of $e^x$, for e any number, "$M(e)e^x$" where M(e) is a number depending on e only, not on x.

It is not too difficult to show that M(2) is less than one and that M(3) is larger than 1 (by numerically approximating the limits). "Choosing" M(e) to be 1 is really choosing a specific value of e such that M(e)= 1. My point about M(2) and M(3) is that there is such a value of e, between 2 and 3. You could, then, by a succesion of numerical approximations, show that M(2.7) is less than 1 but that M(2.8) is greater than 1 so "e" is between 2.7 and 2.8. Or that M(2.71) is less than 1 but that M(2.72) is greater than one so that "e" is between 2.71 and 2.72, etc.

Choosing "e" to be the number such that M(e)= 1 means that the derivative of the function $y= e^x$ is just $e^x$ again.

Oh! Thanks hallsofivy.(= I was rather taken aback by the fact that M(e)=1 and not any other real numbers between probably 2.71 to 2.72 . once again thanks!

Hi icystrike ;
M(e)=1 since e is defined to be the real number such that the area under the function
f(x)=1/x and x=1 ,x=e and the x-axis is equal to 1.
Best Wishes

## What are derivatives?

Derivatives are mathematical tools used to measure the rate of change of a function at a specific point. They are also known as the slope or gradient of a function.

## What is the relationship between derivatives and exponential functions?

Exponential functions are a type of function that can be differentiated using the rules of derivatives. The derivative of an exponential function is equal to the original function multiplied by the natural logarithm of the base of the function.

## How are derivatives and exponential functions used in science?

Both derivatives and exponential functions are used in various fields of science, such as physics, biology, and economics. They can be used to model and analyze real-world phenomena, such as population growth, radioactive decay, and chemical reactions.

## What is the chain rule in derivatives?

The chain rule is a rule used to find the derivative of a composite function, where one function is nested inside another. It states that the derivative of the outer function multiplied by the derivative of the inner function gives the derivative of the composite function.

## Can exponential functions have negative exponents?

Yes, exponential functions can have negative exponents. This results in a fraction with a numerator of 1 and a denominator equal to the original function raised to the absolute value of the exponent. These types of functions are known as rational exponential functions.