Derivative of e^-6x

Learn what is the derivative of e^-6x with formula. Also understand what is the derivative of e to a number a with a step-by-step proof.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of e-6x

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^-6 by applying the first principle of differentiation. In this article, you will learn what the derivative of e-6x is and how to calculate the derivative of e-6x by using different approaches.

What is the derivative of e^-6x?

The derivative of e^(-6x) with respect to x is -6e^-6x. This can also be represented as d/dx (e-6x). The derivative is a measure of how quickly the exponential function e^-6x changes as x changes.

It is important to note that the derivative of e^-6x is always equal to the function itself. Understanding the derivative e^-6x is key in solving problems involving exponential functions and their rates of change.

Derivative of e-6x formula

The derivative formula for e^-6x is;

d/dx (e-6x) = -6e-6x

This means that when taking the derivative with respect to x, the result is equal to the exponential function e^-6x multiplied by -6. Understanding this formula is important for solving problems involving the rate of change of exponential functions.

How do you prove the derivative of e^(-6x)?

There are various ways to derive derivatives of e-6x. Some of these are;

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

Each method provides a different way to compute the e^(-6x) derivative. By using these methods, we can mathematically prove the formula for finding the derivative of e^(-6x).

Derivative e^(-6x) by first principle

According to the first principle of derivative, the ln e^-6x derivative is equal to -6e^-6x. 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-6x by first principle

To prove the derivative of e by using first principle, replace f(x) by e^-6x or replace by e^-x to calculate the derivative of e^-x.


f'(x) = lim e^-6(x+h) - e^-6x/h


f'(x) = lim e^-6x.e^-6h - e^-6x/h

Taking e-6x common as;

f'(x) = lim e^-6x(e^-6h - 1)/h

More simplification,

f'(x) = -e^-6x .lim (e^-6h - 1)/-h

When h approaches to zero,

f'(x) = -6e^-6x lim (e^0 - 1)/-6h

f'(x) = -6e^-6x f(0)


f'(x) = -6e^-6x

Derivative of e^-6x by product rule

Another method to find the derivative e^-6x 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:

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 derivative of e^-6x by product rule

To prove the derivative of e to -6x by using the product rule, we start by assuming that,

f(x) = 1. e^-6x

By using the product rule of differentiation,

f(x) = (e^-6x)'. 1 + (1)'e^-6x

We get,

f(x) = -6e^-6x + 0


f(x) = - 6e^-6x

Derivative of e^-6x using quotient rule

Since the exponential function can be written as its reciprocal. Therefore, the derivative of e^-6x 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 of e-6x by quotient rule

To prove the derivative of e^-6x, we can write it,

f(x) = e^-6x /1 = u/v

Supposing that u = e^-6x and v = 1. Now by the quotient rule solver,

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

f'(x) = [d/dx(e^-6x) - e^-6x .d/dx(1)] / (1)2

= [-6e^-6x] / 1

= -6e^-6x

Hence, we have derived the derivative of e^-6x using the quotient rule of differentiation.

How to find the differential of e^-6x 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 a step-by-step way to calculate derivatives using this tool.

  1. Write the function as e-6x 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-6x.
  2. Now, select the variable by which you want to differentiate e-6x. Here you have to choose x.
  3. Select how many times you want to differentiate e to the -6x. 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 e-6x within a few seconds.

After completing these steps, you will receive the differential of e-6x within seconds. Using online tools can make calculating derivatives much easier and faster, especially for complex functions.

Frequently asked questions

What is the derivative of e to a number?

Since e is also a constant, a constant to the power of another constant is, you guessed it, a constant. The derivative of a constant is always 0 , since constants never change.

What is the derivative of e^-6x?

The derivative of e-6x with respect to x is -5e-6x. Mathematically, the derivative of e to the x is written as;

d/dx (e-6x) = -6e-6x

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