Derivative of e^4x

Learn what is the derivative of an exponential function e to the 4x with formula. Also understand how to prove the derivative of e^4x by first principle.

Alan Walker-

Published on 2023-07-12

Introduction to the derivative of e^4x

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 rules of differentiation. Or, we can directly find the derivative of e4 by applying the first principle of differentiation. In this article, you will learn what the derivative of e4 is and how to calculate the derivative of e4x by using different approaches. 

What is the derivative of e^4x?

The e4x derivative is equal to 4e^4x. This can be represented as d/dx (e^4x).

Essentially, the differential of e^4x measures the rate of change of the function; in this case, it is always equal to the negative of the original function. Understanding the derivative of e^(4x) is important in various fields of mathematics and sciences, such as calculus and physics.

Derivative of e4x formula

The formula for the d/dx e^4x is equal to the 4 multiplied by e^(4x). Mathematically,

 $\frac{d}{dx}(e^{4x}) = 4e^{4x}$

It is important to note that the derivative of e^(4x) is not the same as the derivative of e^(x), which is equal to e^(x).

How do you prove the e^4x derivative?

 There are multiple differentiation techniques to derive derivatives of e4x. Therefore, we can prove the derivative of e4x by using;

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

Derivative of e4x by first principle

According to the first principle of derivative, the ln e^(4x) derivative is equal to 4e^(4x). 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.

Proof of derivative of e4x by first principle

To differentiate e^4x by using first principle, replace f(x) by e^4x.


$f’(x) = \lim_{h\to 0}\frac{e^{4(x+h)} – e^{4x}}{h}$


$f’(x) = \lim_{h\to 0} z\frac{e^{4x}.e^{4h} – e^{4x}}{h}$

Taking ex common as;

$f’(x) = \lim_{h \to 0}\frac{e^{4x}(e^{4h} – 1)}{h}$

More simplification, 

$f’(x) = 4e^{4x}.\lim_{h\to 0}\frac{(e^{4h} – 1)}{4h}$

When h approaches to zero, 

$f’(x) = 4e^{4x}\lim_{h\to 0}\frac{(e^0 – 1)}{4h}$

$f’(x) = 4e^{4x} f’(0)$


$f’(x) = 4e^{4x}$

Derivative of e4x 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 e4x by product rule 

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

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

By using product rule of differentiation,

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

We get,

$f’(x) = 2e^{4x} + 2e^{4x}$

Hence, we have derive the derivative of e^4x by using product rule of two functions.

$f’(x) = 4e^{4x}{2}nbsp;

Derivative of e4x using quotient rule

Another way of finding the e4x derivative 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 e4x by quotient rule 

To prove the derivative of e^4x, we can write it,

$f(x) = \frac{e^{4x}}{1} = \frac{u}{v}$

Supposing that u = e4x and v = 1. Now by quotient rule,

$f’(x) = \frac{(vu’ – uv’)}{v^2}$

$f'(x) = \frac{\frac{d}{dx}(e^{4x}) – e^{4x} .\frac{d}{dx}(1)}{(1)^2}$

$f'(x)= \frac{4e^{4x}}{1}$

$f'(x)= 4e^{4x}{2}nbsp;

Hence, we have derived the derivative of e4x using the quotient rule of differentiation. We can also use the quotient calculator that evaluates the derivative of a quotient function more easily and accurately then manual calculations.

How to find the derivative of e4x with a calculator?

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

  1. Write the function as e4x 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 e4x.
  2. Now, select the variable by which you want to differentiate e^4x. Here you have to choose ‘x’.
  3. Select how many times you want to differentiate e4x 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 e4x within a few seconds.


What is the derivative of e^4x?

The derivative of e4x with respect to x is 4e4x. Mathematically, the derivative of e squared to the x is written as;

 $\frac{d}{dx}(e^{4x}) = 4e^{4x}$

Does every functions have derivative?

This limit may not exist. Therefore It is not necessary that every function has a derivative at every point. A function with a derivative at x=a is said to be differentiable at x=a.

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