# Derivative of e^9x

Learn what is the derivative of an exponential function e to the 9x with formula. Also understand how to prove the derivative of e^9x by first principle.

Alan Walker-

Published on 2023-05-26

## Introduction to the Derivative of e^9x

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^9 by applying the first principle of differentiation. In this article, you will learn what the derivative of e9 is and how to calculate the derivative of e9x by using different approaches.

## What is the derivative of e^(9x)?

The derivative of e^9x is equal to 9e^9x. In other words, the rate of change of the exponential function e to the power of 9x with respect to the variable x is always equal to 9 times e to the power of 9x. This derivative is denoted by d/dx (e9x).

Understanding the product rule and quotient rule proofs for finding the derivative e^9x can help further develop your understanding of this essential calculus concept

## Derivative of e9x formula

The formula for finding the derivative of e raised to the power of 9x is as follows:

d/dx (e^9x) = 9e^9x

This formula expresses the rate of change of the exponential function, indicating that the derivative of e^9x with respect to x is always equal to 9 times e^9x.

## How do you prove the derivative of e9x?

There are different ways to derive derivatives of e9x. Some of these are;

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

Each method provides a different way to compute the e^9 derivative. By using these methods, we can mathematically prove the formula for finding the derivative of e^9.

## Derivative of e9x by first principle

According to the first principle of derivative, the ln e^9x derivative is equal to 9e^9x. 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 e9x by first principle

To prove the derivative of e to the 9x by using first principle, we start by replacing f(x) by e^9x.

f'(x)=limh→0f(x+h)-f(x)/h

f'(x) = lim e^9(x+h) - e^9x/h

Moreover,

f'(x) = lim e^9x.e^9h - e^9x/h

Taking e^9x common as;

f'(x) = lim e^9x(e^9h - 1)/h

More simplification,

f'(x) = 9e^9x .lim (e^9h - 1)/9h

When h approaches to zero,

f'(x) = 9e^9x lim (e^0 - 1)/9h

f'(x) = 9e^9x f(0)

Therefore,

f'(x) = 9e^9x

Moreover, the derivative graph calculator can be use to draw the graph of exponential function.

## Derivative of e^9x by product rule

Another method to find the derivative e^9x 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 e9x by product rule

To prove the derivative of e by using product rule, we start by assuming that,

f(x) = e5x. e4x

By using product rule of differentiation,

f(x) = (e5x). e4x + (e4x)e5x

We get,

f(x) = 5e9x + 4e9x

Hence,

f(x) = 9e9x

## Derivative of e9x using quotient rule

Since the exponential function can be written as the reciprocal of the negative power of e^9x. Therefore, the derivative of cot 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 e^9x derivative by quotient rule

To prove the derivative of e^9x, we can start by writing it,

f(x) = e9x /1 = u/v

Supposing that u = e9x and v = 1. Now by quotient rule,

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

f(x) = [d/dx(e9x) - e9x .d/dx(1)] / (1)2

= [9e9x] / 1

= 9e9x

Hence, we have derived the derivative of e9x using the quotient rule of differentiation. Also, use our quotient rule calculator to evaluate derivative of a quotient function.

## How to find the derivative of e^(9x) 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 you a step-by-step way to calculate derivatives by using this tool.

1. Write the function as e9x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the e^9x derivative.
2. Now, select the variable by which you want to differentiate e9x. Here you have to choose x.
3. Select how many times you want to differentiate e9x to the x. In this step, you can choose 2 for second, 3 for third derivative and so on.
4. Click on the calculate button.

After completing these steps, you will receive the differential of e^9x within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

### What is the most basic exponential function?

A basic exponential function is defined as f(x) = bx, where b is a constant and x is a variable. f(x) = ex is a popular exponential function, where e is Euler's number and e = 2.718.

### What is the derivative of e^9x?

The derivative of e9x with respect to x is 9e9x. Mathematically, the derivative of e to the x is written as;

d/dx (e9x) = 9e9x