## Introduction to the Derivative of e^7x

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 e7 by applying the first principle of differentiation.

In this article, you will learn what the derivative of e7x is and how to calculate the derivative of e7x by using different approaches.

## What is the derivative of e^7x?

**The derivative of e^7 with respect to the variable x is 7e^7x. This is commonly denoted as d/dx (e^(7x)). The derivative of the exponential function e^7x represents the rate of change of the function, which is always equal to the exponential function itself.**

Knowing the e^7x derivative is important in calculus, as it is a fundamental concept used in various mathematical models and applications.

## Derivative of e^(7x) formula

The formula to calculate derivative of e^7 is:

d/dx (e^7x) = 7e^7x

This formula gives the rate of change of the exponential function e^7x with respect to the variable x, and it can be used to differentiate any function that involves the exponential constant e raised to a multiple of 7x. By using this formula, you can easily solve calculus problems that require finding the derivative of functions involving e^7x.

## How do you prove the derivative of e7x?

There are different ways to derive derivatives of e7x. Three common methods are;

- First Principle
- Product Rule
- Quotient Rule

Each method provides a different way to compute the e7x derivative. By using these methods, we can mathematically prove the formula for finding the derivative of e^7x

## Derivative of e7x by first principle

According to the first principle of derivative, the ln e^7x derivative is equal to 7e^7x. The derivative of a function by 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 limit definition of derivative.

## Proof of derivative of e7x by first principle

To prove the derivative of e^(7x) by using first principle, we start by replacing f(x) by e.

f'(x)=limh→0f(x+h)-f(x)/h

f'(x) = lim e7(x+h) - e7x/h

Moreover,

f'(x) = lim e7x.e7h - e7x/h

Taking e7x common as;

f'(x) = lim e7x(e7h - 1)/h

More simplification,

f'(x) = 7e7x .lim (e7h - 1)/7h

When h approaches to zero,

f'(x) = 7e7x lim (e0 - 1)/7h

f'(x) = 7e7x f(0)

Therefore,

f'(x) = 7e7x

The derivative of e can also be calculated by using first principle of differentiation.

## Derivative of e7x by product rule

Another method to find the derivative e^7x 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 e7x by product rule

To prove the derivative of e^(7x) by using product rule, assume that,

f(x) = e3x. e4x

By using product rule of differentiation,

f(x) = (e3x). e4x + (e3x)e4x

We get,

f(x) = 3e7x + 4e7x

Hence,

f(x) = 7e7x

## Derivative of e7x using quotient rule

Since the exponential function can be written as the reciprocal of 1 and e-7x. Therefore, the derivative of e7x 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 e7x by quotient rule

To prove the derivative of e^7x, we can start by writing it,

f(x) = e7x /1 = u/v

Supposing that u = e7x and v = 1. Now by quotient rule,

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

f(x) = [d/dx(e7x) - e7x .d/dx(1)] / (1)2

= [7e7x] / 1

= 7e7x

Hence, we have derived the derivative of e7x using the quotient rule of differentiation.

## How to find the derivative of e7x with a calculator?

The easiest way to calculate the e^7x derivative is by using an online derivative calculator. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

- Write the function as e7x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the differentiate e^7x.
- Now, select the variable by which you want to differentiate e7x. Here you have to choose x.
- Select how many times you want to differentiate e to the 7x. In this step, you can choose 2 for second, 3 for third derivative and so on.
- Click on the calculate button. After this step, you will get the derivative of e7x within a few seconds.

## Frequently asked questions

### What is the derivative of e^7x?

The derivative of e7x with respect to x is 6e7x. Mathematically, the derivative of e to the x is written as;

d/dx (e7x) = 7e7x

### What is the rule of exponential function?

An exponential function is a mathematical function in the form of f(x) = ex, where x is the variable and e is a constant which is called the base of the function. The value of e is always greater than zero.