Derivative of e^x^2

Learn what is the derivative of e to the squared x and its formula. Also understand how to prove the derivative of e^x^2 by using chain rule and product rule.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of e^x^2

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 ex2 by applying the first principle of differentiation. In this article, you will learn what the derivative of e^x squared is and how to calculate the derivative of ex2 by using different approaches.

What is the derivative of e^x^2?

The derivative of e^x^2, denoted by d/dx (e^x^2), is equal to 2xe^x2. This tells us the rate of change of the exponential function e with respect to the variable x. Essentially, the derivative measures how much the function changes as we make small adjustments to x.

Derivative of ex2 formula

The formula for the e^x^2 differentiation is equal to 2xe^x^2, which means that the differentiation of e^x^2 with respect to the variable x is equal to twice the value of x multiplied by e raised to the power of x^2. Mathematically, 

$\frac{d}{dx}(e^{x^2}) = 2xe^{x^2}$

How do you prove the derivative of ex2?

There are multiple ways to derive the derivatives of ex2. Some of these are;

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

Each method provides a different way to compute the e^x^2 derivative. By using these methods, we can mathematically prove the formula for finding the differential of e^x^2. Let's understand how to differentiate e^x^2.

Derivative of ex2 by first principle

According to the first principle of derivative, the e^2x derivative is equal to 2xe^x^2. 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,

$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$

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 to the x^2 by first principle

To prove the e^x^2 differentiation by using the first principle, we start by replacing f(x) by e^x^2.


$f'(x) = \lim_{h\to 0} \frac{e^{(x+h)^2} - e^{x^2}}{h}$


$f'(x) = \lim_{h \to 0} \frac{e^{x^2+h^2+2xh} - e^{x^2}}{h}$

Taking e^x^2 common as;

$f'(x) = \lim_{h\to 0} \frac{e^{x^2}(e^{h^2+2xh} - 1)}{h}$

More simplification

$f'(x) = 2xe^{x^2}.\lim_{h\to 0}\frac{e^{h^2+2xh} - 1}{2xh}$

When h approaches to zero,

$f'(x) = 2xe^{x^2} \lim_{h\to 0}\frac{e0 - 1}{2xh}$

$f'(x) = 2xe^{x^2} f(0)$


$f'(x) = 2xe^{x^2}$

Hence the derivative of e^x^3 can also be calculated by using first principle.

Derivative of ex2 by product rule

Another method to find the derivative e^x^2 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:

$\frac{d}{dx}(uv) = u\frac{dv}{dx} + \frac{du}{dx}v$

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 by product rule

To prove the e^x2 derivative by using the product rule, we start by assuming that,

$f(x) = e^{x^2}.(1)$

By using the product rule of differentiation,

$f(x) = (e^{x^2})' + (1)'e^{x^2}$

We get,

$f'(x) = 2xe^{x^2} + 0$


$f'(x) = 2xe^{x^2}$

Derivative of e to the x2 using quotient rule

Since the cotangent is the reciprocal of the tangent. Therefore, the differential of e^x^2 can also be calculated by using the quotient rule. The quotient rule is defined as;

$\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f'(x).g(x)-g'(x).f(x)}{(g(x))^2}$

Proof of e^x^2 differentiation by quotient rule

To differentiate e^x^2, we can write it,

$f(x) = \frac{e^{x^2}}{1} = \frac{u}{v}$

Supposing that u = ex2 and v = 1. Now by the quotient rule,

$f'(x) = \frac{vu' - uv'}{v^2}$

$f'(x) = \frac{\frac{d}{dx}(e^{x^2}) - e^{x^2}.\frac{d}{dx}(1)}{(1)^2}$

$f'(x)=\frac{2xe^{x^2} - e^{x^2}(0)}{1}$

$f'(x)= \frac{2xe^{x^2}}{1}$

$f'(x)= 2xe^{x^2}$

Hence, we have derived the derivative of e^x^2 using the quotient rule of differentiation.

How to find the differentiation of e^x^2 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 a step-by-step way to calculate derivatives using this tool.

  1. Write the function as e^x^2 in the enter function box. In this step, you need to provide input value as a function as you have to calculate the e^x2 derivative.
  2. Now, select the variable by which you want to differentiate e^x^2. Here you have to choose x.
  3. Select how many times you want to differentiate e to the x^2. In this step, you can choose 2 for the second derivative, 3 for third derivative and so on.
  4. Click on the calculate button. After this step, you will get the derivative e^x^2 within a few seconds.

Frequently asked questions

What is the derivative of e^x2?

The differential of e^x^2 with respect to x is 2xex2. Mathematically, the derivative of e to the squared x is written as;

$\frac{d}{dx}(e^{x^2}) = 2xe^{x^2}$

Who invented e in calculus?

Leonhard Euler, the great mathematician, discovered the number e and calculated its value to 23 decimal places. It is a transcendental number and is often referred to as Euler's number.

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