Why the Derivative of e^x is e^x

The mathematical constant e, approximately 2.71828, has unique properties that make it a fundamental constant in calculus. One of the most intriguing of these properties is the fact that the derivative of the exponential function e^x is e^x itself. This article will explore why this is true, starting from the basic definition of the derivative and proceeding through important mathematical concepts.

The Definition of the Derivative

The derivative of a function f(x) at a point x is defined as:

f'(x) lim_{h to 0} (f(x h) - f(x))/h

Applying the Definition to e^x

Let's apply this definition to the function f(x) e^x:

f(x) lim_{h to 0} (e^{x h} - e^x) / h

Using the properties of exponents, we can rewrite e^{x h} as e^x cdot e^h:

f(x) lim_{h to 0} (e^x cdot e^h - e^x) / h

Factoring out e^x from the numerator, we get:

f(x) e^x cdot lim_{h to 0} (e^h - 1) / h

Evaluating the Limit

The limit lim_{h to 0} (e^h - 1) / h is a well-known limit that equals 1. This can be demonstrated using the Taylor series expansion of e^h, which is:

e^h 1 h h^2/2! h^3/3! ...

When we take the limit as h approaches 0, we find that the first-order term h dominates, and the limit equals 1.

Conclusion

Therefore, we have:

f(x) e^x cdot 1 e^x

Thus, the derivative of e^x is e^x. This property is a direct result of the exponential function with base e having the unique characteristic where its rate of change at any point is equal to its value at that point.

Note: This is a fundamental property of the e^x function, and it plays a crucial role in calculus and many areas of science and engineering.