Proof of Derivative of ex

The derivative of cot x can be proved using the following ways:

• By using First Principle of Derivative
• By using Derivative of a^x

Differentiation of e^x Using First Principle

We are going to prove the differentiation of e^x is e^x by using the first principle of derivatives. We will use some basic rules and formulas of exponential functions and derivatives that are given below.

f'(x) = lim h→0 [f(x + h) – f(x)] / h

Also, we know that,

e^x+h = e^x.e^h

lim _x→0 (e^x – 1) / x = 1

Using above formulas, we get,

d(e^x)/dx = lim _h→0 [e^x+h – e^x] / h

= lim _h→0 [e^x.e^h – e^x] / h

= lim _h→0 e^x [e^h – 1] / h

= e^x lim _h→0 [e^h – 1] / h

= e^x × 1 = e^x

Hence we have proved that the derivative of e^x to be equal to e^x

Differentiation of e^x Using Derivative of a^x

Exponential function are expressed in the form f(x) = a^x , where ‘a’ is a constant (real number) and x is variable.

Derivative of exponential function f(x) = a^x is find by the formula,

f'(x) = (ln a).a^x…(i)

Substituting value a = e in eq (i), we get the differentiation of e^x which is given by

f'(x) = (ln e) e x

= 1 × e^x [As, ln e = 1]

= e^x

Hence, the derivative of e to the power x is e x .

Also Read,

• Differentiation of Trigonometric Function
• Differentiation Formulas
• Derivative of e^2x 

Derivative of e^x is e^x. Derivative of e^x means finding the change in the exponential function with respect to the independent variable. The process of finding the derivative is known as differentiation. The derivative of e^x is e^x. Understanding the derivative of e^x is an important concept in calculus as it offers insights into the dynamic nature of exponential growth.

In this article, we will talk about derivatives of e^x, what is derivative, and its some basic rules.