# 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.

$f′(x)=\lim_{h→0}\frac{f(x+h)−f(x)}{h}$

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

Moreover,

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

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,

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

$f’(x) = 4e^{4x} f’(0)$

Therefore,

$f’(x) = 4e^{4x}$

## 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:

## 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,

$f’(x) = \frac{(vu’ – uv’)}{v^2}$

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

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

### 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.