What is the Derivative of e^3

The derivative of e cube is zero. Note that e cube is written as e³. In this post, we will learn how to find the derivative of e³.

Derivative of e³ Formula

The formula for the derivative of e³ is 0. This formula is written below.

${\displaystyle \frac{d}{dx}}(e^3)=0$ or $(e^3{)}^{\prime }=0$.

Here, the prime ${}^{\prime }$ denotes the first-order derivative.

What is the Derivative of e³?

Answer: The derivative of e³ is 0.

Explanation:

It is known that the value of $e$ is given by the following convergent series:

$e=\sum _{n=0}^{\mathrm{\infty }}$ $=1+{\displaystyle \frac{1}{1!}}+{\displaystyle \frac{1}{2!}}+\cdots$

The number e is irrational, and its value is approximately equal to 2.7182818 (up to 7 decimal places). As e is a fixed number, we conclude that e is a constant.

This implies that e³ is a constant with respect to x.

$\therefore {\displaystyle \frac{d}{dx}}(e^3)=0$ by the rule Derivative of a constant is 0.

Thus, the derivative of e³ is equal to 0.

Also Read: 

Derivative of e²

Derivative of e^2x

Derivative of e^3x

Derivative of log(3x)

Derivative of e³ by First Principle

Let f(x)=e³. Note that e³ is independent of x, so we have f(x+h)=e³ for any values of x and h. By the first principle, the derivative of f(x)=e³ is equal to

${\displaystyle \frac{d}{dx}}(f(x))$ $=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h)-f(x)}{h}}$

So ${\displaystyle \frac{d}{dx}}(e^3)$ $=\underset{h\to 0}{lim}{\displaystyle \frac{e^3-e^3}{h}}$

$=\underset{h\to 0}{lim}{\displaystyle \frac{0}{h}}$

$=\underset{h\to 0}{lim}0$

$=0$.

Hence, the derivative of e³ by the limit definition is equal to 0.

Also Read:

Derivative of log(sin x)

Derivative of log(cos x)

FAQs