## Introduction to the Derivative of e-5x

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-5 by applying the first principle of differentiation. In this article, you will learn what the derivative of e-5x is and how to calculate the derivative of e-5x by using different approaches.

## What is the derivative of e^-5x?

The derivative of the exponential function e^-5x with respect to x is -5e-5x. This can also be represented as d/dx (e-5x) Essentially, this gives us the rate of change of the function, which is always equal to the function itself. Knowing how to differentiate e^-5x is important in solving problems related to growth and decay processes in various fields such as finance, physics, and chemistry.

## Derivative of e-5x formula

The formula for finding the derivative of e^(-5x) is -5e^-5x. This means that if you take the derivative of e^-5x with respect to x, the result will be -5 times e^-5x. Mathematically, it can be written as;

d/dx (e-5x) = -5e-5x

By understanding and applying this formula correctly, you can solve problems related to exponential functions and their behavior.

## How do you prove the derivative of e-5x?

There are different ways to derive derivatives of e-5x. Some of these are;

- First Principle
- Product Rule
- Quotient Rule

Each method provides a different way to compute the e^(-5x) derivative. By using these methods, we can mathematically prove the formula for finding the derivative of e^(-5x).

## Derivative of e-5x by first principle

According to the first principle of derivative, the ln e^-5x derivative is equal to -5e^-5x. 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-5x by first principle

To prove the derivative of e by using the first principle, we start by replacing f(x) with e^-5 or replace by e^-4x to find the derivative of e^-4x.

f'(x)=limh→0f(x+h)-f(x)/h

f'(x) = lim e-5(x+h) - e-5x/h

Moreover,

f'(x) = lim e-5x.e-5h - e-5x/h

Taking e-5x common as;

f'(x) = lim e-5x(e-5h - 1)/h

More simplification,

f'(x) = -e-5x .lim (e-5h - 1)/-h

When h approaches to zero,

f'(x) = -5e-5x lim (e0 - 1)/-5h

f'(x) = -5e-5x f(0)

Therefore,

f'(x) = -5e-5x

## Derivative of e^-5x by product rule

Another method to find the e^-5x 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 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 derivative of e-5x by product rule

To prove the derivative of e to the -5x by using the product rule, we start by assuming that,

f(x) = 1. e-5x

By using the product rule of differentiation,

f(x) = (e-5x). 1 + (1)e-5x

We get,

f(x) = -5e-5x + 0

Hence,

f(x) = - 5e-5x

## Differentiation of e^-5x using quotient rule

Since the exponential function can be written in a fractional form. Therefore, the derivative of e^-5x can also be calculated by using the quotient law calculator. 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-5x by quotient rule

To prove the e^-5x derivative, we can start by writing it,

f(x) = e-5x /1 = u/v

Supposing that u = e-5x and v = 1. Now by the quotient rule,

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

f(x) = [d/dx(e-5x) - e-5x .d/dx(1)] / (1)2

= [-5e-5x] / 1

= -5e-5x

Hence, we have derived the derivative of e-5x using the quotient rule of differentiation.

## How to find the e^-5x derivative with a calculator?

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

- Write the function as e-5x 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^-5x.
- Now, select the variable by which you want to differentiate e^-5x. Here you have to choose x.
- Select how many times you want to differentiate e to the -5x. In this step, you can choose 2 for the second, 3 for third derivative and so on.
- Click on the calculate button. After this step, you will get the derivative of e-5x within a few seconds.

After completing these steps, you will receive the differential of e^(-5x) within seconds. Using online tools can make calculating derivatives much easier and faster, especially for complex functions.

You can also use nth derivative calculate to compute the derivative of the function up to n times.

## Frequently asked questions

### What are four types of exponent?

Exponents are classified into four types: positive, negative, zero, and rational or fractional. The value of the number can be interpreted by interpreting the exponent as the total number of times the base number must be multiplied by the same base.

### What is the derivative of e^-5x?

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

d/dx (e-5x) = -5e-5x