Derivative of e^x^3

Learn what is the derivative of e to the x cubic and its formula. Also understand how to prove the derivative of e^x^3 by using derivative rules.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of ex3

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 ex3 by applying the first principle of differentiation. In this article, you will learn what the derivative of e squared is and how to calculate the e^x^3 derivative by using different approaches.

What is the derivative of e to the x3?

The derivative of e^x^3, denoted as d/dx (ex3) is equal to 3x^2.e^x^3. It  is a fundamental concept in calculus.

Simply put, it represents the rate of change of the exponential function e with respect to the variable x, which is always equal to the exponential function itself. In other words, the derivative of e^(x^3) is e^(x^3), making it an important function in many applications in science and engineering. 

Derivative of ex3 formula

The formula for finding the derivative of e^(x^3) is e^(x^3) multiplied by the derivative of the exponent, which is 3x^2. Mathematically it can be written as;

$\frac{d}{dx}(e^x^3) = 3x^2e^{x^3}$

How do you prove the derivative of ex3?

There are various methods to derive derivatives of ex3. Three common methods are;

  1. Product Rule
  2. Quotient Rule
  3. Chain Rule

Each method provides a different way to compute the e^x^3 derivative. By using these methods, we can mathematically prove the formula for finding the derivative of e^x^3.

Derivative of ex3 by product rule

Another method to find the derivative e^x^3 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 of differentiation 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 by product rule

To prove the e^x^3 differentiation by using product rule, we start by assuming that,

f(x) = ex3. (1)

By using the product rule formula,

f(x) = (ex3) + (1)ex

We get,

f(x) = 2xex3 + 0


f(x) = 2xex3

Derivative of e^x^3 using quotient rule

Although the differentiation of e^x^3 is typically calculated using the formula d/dx (e^(x^3)) = 3x^2e^(x^3). It is also possible to use the quotient rule to find the derivative. The quotient rule in differentiation to find the derivative of two functions f(x) and g(x) is equal to,

d/dx (f/g) = f(x). g(x) -g(x).f(x) /{g(x)}2

This alternative method can be helpful in certain situations where using the quotient rule may be more convenient than the direct formula.

Proof of derivative of ex3 by quotient rule

To prove the derivative of e^x3, we can start by writing it,

f(x) = ex3 /1 = u/v

Supposing that u = ex3 and v = 1. Now by quotient rule calculator,

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

f(x) = [d/dx(ex3) - ex3. d/dx(1)] / (1)2

= [2xex3 - ex3(0)] / 1

= 2xex3 / 1

= 2xex3

Hence, we have derived the derivative of ex3 using the quotient rule of differentiation.

Derivative of e^x^3 by chain rule

To calculate the derivative of ex3, we can use the chain rule since the function ex3 can be expressed as a combination of two functions. The chain rule of derivatives states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. The chain rule of derivative is defined as;

dy / dx = dy / du x du / dx

Proof of differentiation of e^x^3 by chain rule

To prove the derivative of ex3 by using chain rule, we start by assuming that,

y = ex3

It can be written as;

y = ef(x)

By using chain rule of differentiation calculator,

y = f(x)ef(x)


f(x) = 3x2

Now we have,

y = 3x2ex3

Hence we have verified the derivative e^x^3.

How to find the e^x^3 differentiation 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 you a step-by-step way to calculate derivatives by using this tool.

  1. Write the function as ex3 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^x^3.
  2. Now, select the variable by which you want to differentiate ex3. Here you have to choose x.
  3. Select how many times you want to differentiate ex3 to the x. In this step, you can choose 2 for second, 3 for third derivative calculator and so on.
  4. Click on the calculate button.

After completing these steps, you will receive the e^x^3 differentiation 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^x3?

The derivative of ex3 with respect to x is 2xex3. Mathematically, the derivative of e to the squared x is written as;

d/dx (ex3) = 2xex3

What is the derivative of e constant?

It is an exponential function but it is a constant. Moreover, the power of a constant is also considered as a constant. And we know that the derivative of a constant is zero. Therefore, the derivative of a constant function will never change.

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