**Introduction to the Derivative of e^ax**

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

**What is the derivative of e^(ax)?**

The derivative of e^ax is equal to ae^ax. This can be represented as d/dx (e^ax). Essentially, the differential of e^ax measures the rate of change of the function; in this case, it is always equal to the negative of the original function. Understanding the derivative of e^(ax) is important in various fields of mathematics and sciences, such as calculus and physics.

**Derivative of e^ax formula**

The formula for the differentiation of e^(ax) is equal to the a multiplied by e^(ax). Mathematically,

d/dx (e^ax) = ae^ax

It is important to note that the derivative of e^(ax) is not the same as the derivative of e^(x), which is equal to e^(x).

**How do you prove the derivative of eax?**

There are several derivative techniques for finding the derivative of e^(ax). The most common ways are;

- First Principle
- Product Rule
- Quotient Rule

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

**Derivative of e to the ax by first principle**

According to the first principle of derivative, the ln e^(ax) derivative is equal to ae^(ax). 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 calculator.

**Proof of derivative of e^a by first principle**

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

f'(x)=limh→0f(x+h)-f(x)/h

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

Moreover,

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

Taking e^ax common as;

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

More simplification,

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

When h approaches to zero,

f'(x) = ae^ax lim (e0- 1)/2h

f'(x) = ae^ax f(0)

Therefore,

f'(x) = ae^ax

Hence the derivative of e^ax can also be calculated by using first principle. This method is also applicable to calculate te derivative of e^x^2.

**Differentiation of e^ax by product rule**

Another method to find the derivative e^(ax) 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 product rule formula for a product of two funcitons 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^ax by product rule**

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

f(x) = 1. e^ax

By using product rule differentiation calculator,

f'(x) = (e^ax). 1 + (1)e^ax

We get,

f'(x) = ae^ax + 0

Hence,

f'(x) = ae^ax

**Derivative of e^ax using quotient rule**

Since the exponential function can be written as the reciprocal of e^(ax). Therefore, the derivative e^ax 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 differentiation of e^ax by quotient rule**

To prove the e^ax derivative, we can start by writing it,

f(x) = e^ax /1 = u/v

Supposing that u = e^ax and v = 1. Now by quotient rule calculator,

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

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

= [ae^ax] / 1

= ae^ax

Hence, we have derived the differential of e^ax using the quotient rule of differentiation. Also use our quotient rule calculator to calculate rate of change of any exponential function by using quotient rule.

**How to find the differentiation of e^ax with a calculator?**

The easiest way to calculate the derivative of e to the ax 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.

- Write the function as e^ax 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^ax.
- Now, select the variable by which you want to differentiate eax. Here you have to choose x.
- Select how many times you want to differentiate e to the ax. In this step, you can choose 2 for the second derivative, 3 for the third derivative and so on.
- Click on the calculate button. After this step, you will get the derivative of e2x within a few seconds.

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

**Frequently asked questions**

**What is the derivative of e to the power ax?**

The derivative of e to the power ax is equal to the product of a and e^ax. It is denoted by d/dx(e^ax). Mathematically, it can be written as

d/dx(e^ax)=ae^ax

**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 5-x is written as;

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