Derivative of e^3x

Learn what is the derivative of e cubic x and its formula. Also understand how to prove the derivative of e^3x by using implicit differentiation.

Alan Walker-

Published on 2023-07-12

Introduction to the derivative of e^3x

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of e can be calculated by following the differentiation rules. Or, we can directly find the derivative of e3 by applying the first principle of differentiation. In this article, you will learn what the derivative of e cubic is and how to calculate the derivative of e3x by using different approaches. 

What is the derivative of e^3x?

The derivative of e to the x with respect to the variable ‘x’ is equal to 3e^3x. It is denoted by d/dx (e^3x). It is the rate of change of the exponential function e and it is always equal to the exponential function itself. 

Derivative of e3x formula

The formula for the differentiation of e^(3x) is equal to the 3 multiplied by e^(3x). Mathematically,

 $\frac{d}{dx}(e^{3x}) = 3e^{3x}$

It is important to note that the derivative of e^(3x) is not the same as the derivative of e^(x), which is equal to e^(x).

How do you prove the derivative of e3x?

 There are numerous ways to derive derivatives of e^3x. Therefore, we can prove the derivative of e3x by using;

  1. First Principle
  2. Product Rule
  3. Quotient Rule

Derivative of e3x by first principle

According to the first principle of derivative, the ln e^(3x) derivative is equal to 3e^(3x). The derivative of a function by the first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

$f’(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$

This formula allows us to determine the rate of change of a function at a specific point by using the limit definition of the derivative. This formula is also used in derivative definition calculator that calculates derivative of a function quickly.

Proof of derivative of e3x by first principle

To prove the derivative of e^3x by using first principle, replace f(x) by e.


$f’(x) =\lim_{h\to 0}\frac{e^{3(x+h)} – e^{3x}}{h}$


$f’(x) =\lim_{h\to 0}\frac{e^{3x}.e^{3h} – e^{3x}}{h}$

Taking ex common as;

$f’(x) =\lim_{h\to 0}\frac{e^{3x}(e^{3h} – 1)}{h}$

More simplification, 

$f’(x) = 3e^{3x}.\lim_{h\to 0}\frac{(e^{3h} – 1)}{3h}$

When h approaches to zero, 

$f’(x) = 3e^{3x}\lim_{h\to 0}\frac{(e^0 – 1)}{3h}$

$f’(x) = 3e^{3x} f’(0)$


$f’(x) = 3e^{3x}$

Derivative of e3x by product rule

Another method to find the derivative e^(4x) is the product rule formula which is used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the product rule calculator is:

$\frac{d}{dx}(uv) = u(\frac{dv}{dx}) + (\frac{du}{dx})v{2}nbsp;

In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x.

Proof of derivative of e3x by product rule 

To prove the derivative of e by using product rule, we start by assuming that,

$f(x) = e^{2x}. e^x$

By using product rule of differentiation,

$f’(x) = (e^{2x})’. e^x + (e^x)’e^{2x}$

We get,

$f’(x) = 2e^{3x} + e^{3x}


$f’(x) = 3e^{3x}{2}nbsp;

Derivative of e3x using quotient rule

Another way of finding the derivative of e^4x is the quotient rule formula. This method is used when we have to deal with a fraction of two functions. The quotient rule is defined as;

$\frac{d}{dx}(\frac{f}{g}) = \frac{f(x). g’(x) –g(x).f’(x)}{(g(x))^2}$

Proof of derivative of e3x by quotient rule 

 To prove the derivative of e^3x, we can write it,

$f(x) = \frac{e^{3x}}{1} = \frac{u}{v}$

Supposing that u = e3x and v = 1. Now by quotient rule,

$f’(x) = \frac{(vu’ – uv’)}{v^2}$

$f'(x) = \frac{\frac{d}{dx}(e^{3x}) – e^{3x} .\frac{d}{dx}(1)}{(1)^2}$

$f'(x)= \frac{3e^{3x}}{1}$

$f'(x)= 3e^{3x}{2}nbsp;

Hence, we have derived the derivative of e3x using the quotient rule of differentiation calculator.

How to find the derivative of e3x with a calculator?

The easiest way to calculate the derivative of e is by using an online tool. You can use our differentiation calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

  1. Write the function as e3x in the “enter function” box. In this step, you need to provide input value as a function as you have to calculate the derivative of e3x.
  2. Now, select the variable by which you want to differentiate e3x. Here you have to choose ‘x’.
  3. Select how many times you want to differentiate e3x to the x. In this step, you can choose 2 for second, 3 for third derivative and so on.
  4. Click on the calculate button. After this step, you will get the derivative of e3x within a few seconds.


What is the derivative of e^3x?

The derivative of e3x with respect to x is 3e3x. Mathematically, the derivative of e squared to the x is written as;

$\frac{d}{dx}(e^{3x}) = 3e^{3x}$

Are exponential functions differentiable everywhere?

Exponential functions are always continuous. It is because they are always differentiable and continuity is a necessary but it is not sufficient condition for differentiability.

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