Derivative of e^-4x

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

Alan Walker-

Published on 2023-07-10

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 e-4 by applying the first principle of differentiation. In this article, you will learn what the derivative of e-4x is and how to calculate the derivative of e-4x by using different approaches.

What is the derivative of e^-4x?

The e^-4x derivative with respect to x is -4e-4x. This is also represented as d/dx (e-4x ). The derivative measures the rate of change of the exponential function e^-4x, which always equals the function itself. Knowing the derivative of e^-4x is useful in various fields such as calculus and physics

Derivative of e-4x formula

The derivative of e^(-4) can be found using the formula;

d/dx (e-4x ) = -4e-4x

This formula shows that the derivative of e^(-4x) is equal to the exponential function e. Understanding this formula is important in calculus and related fields, where it's used to solve problems related to exponential functions.

How do you prove the derivative of e-4x ?

There are multiple methods to derive the derivative of e-4x, including;

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

Each method provides a different way to compute the e-4x derivative. By using these methods, we can mathematically prove the formula for finding the derivative of e-4x. 

Related: Learn how to calculate the derivative of e^4x online.

Derivative of e-4x 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 calculator 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 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 e-4x by first principle

To prove the derivative of e using the first principle, we start by replacing f(x) with e^-4x.


f'(x) = lim e-4(x+h) - e-4x /h


f'(x) = lim e-4x .e-4h - e-4x /h

Taking e-4x common as;

f'(x) = lim e-4x (e-4h - 1)/h

More simplification,

f'(x) = -e-4x .lim (e-4h - 1)/-h

When h approaches to zero,

f'(x) = -4e-4x lim (e0 - 1)/-4h

f'(x) = -4e-4x f(0)


f'(x) = -4e-4x

You can also calculate the derivative of e-4x by using the derivative graph calculator.

Derivative of e^(-4x) by product rule

Another method to find the e^4x derivative 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 derivative product rule calculator is:

d/dx(uv) = u(dv/dx) + (du/dx)v 

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 e^(-4x) by product rule

To differentiate e^-4x by using the product rule, we start by assuming that,

f(x) = 1. e-4x

By using the product rule of differentiation,

f(x) = (e-4x ). 1 + (1)e-4x

We get,

f(x) = -4e-4x + 0


f(x) = - 4e-4x

Derivative of e^(-4x) using quotient rule

Since the exponential function e^-4x can be written as a fraction, therefore, the derivative of e^-4x can also be calculated by using the quotient rule. The quotient rule is defined as;

d/dx (f/g) = f(x). g(x) -g(x).f(x) /{g(x)}2

Proof of derivative e^-4x by quotient rule

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

f(x) = e-4x /1 = u/v

Supposing that u = e-4x and v = 1. Now by quotient rule calculator,

f(x) = (vu - uv)/v2

f(x) = [d/dx(e-4x ) - e-4x .d/dx(1)] / (1)2

= [-4e-4x ] / 1

= -4e-4x

Hence, we have derived the derivative of e-4x using the quotient rule of differentiation. Also, the quotient rule can be used to solve the differentiation of e^x^2.

How to find the derivative of e^-4x with a calculator?

The easiest way to calculate the derivative of e^(-4x) is by using an online tool. You can use our derivative calculator for this. Here, we provide a step-by-step way to calculate derivatives using this tool.

  1. Write the function as e-4x 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 e-4x.
  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 e to the -4x. In this step, you can choose 2 for the second derivative, 3 for the third derivative and so on.
  4. Click on the calculate button. After this step, you will get the derivative of e-4x within a few seconds.

After completing these steps, you will receive the e^-4x derivative within seconds. Using online tools can make calculating derivatives much easier and faster, especially for complex functions.

Frequently asked questions

Why is the derivative of the exponential function the same?

An exponential function derivative is a constant multiplied by itself. We can see from this definition that the function has the following truly remarkable property. As a result, it is its own derivative. In other words, the slope of the plot is equal to its height, or its second coordinate.

What is the derivative of e^-4x?

The derivative of e^(-4x) with respect to x is -4e^(-4x) . Mathematically, the derivative of e to the x is written as;

d/dx (e^(-4x) ) = -4e^(-4x)

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