Derivative of e^-3x

Learn what is the derivative of an exponential function e to the x with formula. Also understand how to prove the derivative of e^-3x by first principle.

Alan Walker-

Published on 2023-05-26

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 rules of differentiation. Or, we can directly find the derivative of e^-3 by applying the first principle of differentiation. In this article, you will learn what the derivative of e-3x is and how to calculate the derivative of e-3x by using different approaches.

What is the derivative of e^(-3x)?

The derivative of e to the negative cube x with respect to x is -3e^-3x. This can be represented by d/dx (e-3x). Essentially, the derivative gives the rate of change of the exponential function e. It's worth noting that the derivative of e^-3x is always equal to the exponential function itself. Understanding the derivative of e^-3x is important in calculus and related fields, where it's used to solve various problems involving exponential functions.

Derivative of e-3x formula

The derivative of e cubed to the x with respect to x can be found using the formula;

d/dx (e-3x) = -3e-3x

This formula shows that the derivative of e^-3x 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-3x?

There are various methods to derive the derivative of e-3x, including;

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

Each method provides a different way to compute the e^(-3x) derivative. By using these methods, we can mathematically prove the formula for finding the derivative of e^(-3x).

Derivative of e-3x 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 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-3x by first principle

To prove the derivative of e by using first principle, we start by replacing f(x) by e^(-3x).


f'(x) = lim e-3(x+h) - e-3x/h


f'(x) = lim e-3x.e-3h - e-3x/h

Taking e-3x common as;

f'(x) = lim e-3x(e-3h - 1)/h

More simplification,

f'(x) = -e-3x .lim (e-3h - 1)/-h

When h approaches to zero,

f'(x) = -3e-3x lim (e0 - 1)/-3h

f'(x) = -3e-3x f(0)


f'(x) = -3e-3x

Similarly, the derivative of e-4x can be calculated by using first principle.

Derivative of e^(-3x) by product rule

Another method to find the derivative e^-3x 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 derivative 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^-3x derivative by product rule

Todifferentiate e^-3x by using the product rule, we start by assuming that,

f(x) = 1. e-3x

By using product rule of differentiation,

f(x) = (e-3x). 1 + (1)e-3x

We get,

f(x) = -3e-3x + 0


f(x) = - 3e-3x

Derivative of e^-3x using quotient rule

Since the exponential function can be written as the reciprocal of e^3x. Therefore, the derivative of e^-3x 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^(-3x) by quotient rule

To prove the derivative of e^-3x, we can start by writing it,

f(x) = e-3x /1 = u/v

Supposing that u = e-3x and v = 1. Now by the quotient rule,

f(x) = (vu - uv)/e2

f(x) = [d/dx(e-3x) - e-3x .d/dx(1)] / (1)2

= [-3e-3x] / 1

= -3e-3x

Hence, we have derived the derivative of e-3x using the quotient rule of differentiation.

How to find the derivative of e-3x with a calculator?

The easiest way to calculate the derivative of e 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-3x 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-3x.
  2. Now, select the variable by which you want to differentiate e^-3x. Here you have to choose x.
  3. Select how many times you want to differentiate e-3x to the x. In this step, you can choose 2 for the second, 3 for the third derivative and so on.
  4. Click on the calculate button. After this step, you will get the derivative of e-3x within a few seconds.

After completing these steps, you will receive the differential of e^(-3x) within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

Frequently asked questions

What is a negative exponential called?

A negative exponent is defined as the multiplicative inverse of the base raised to the power which is of the opposite sign of the given power. In other words, it is the reciprocal of positive exponential function.

What is the derivative of e^-3x?

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

d/dx (e-3x) = -3e-3x

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