Understanding the Derivative of e^x^2 and Basic Concepts

Understanding the derivative of e^x^2 is a fundamental concept in calculus, particularly when dealing with exponential functions. This article will break down the solution step-by-step and explore related concepts.

What is the Derivative of e^x^2?

The derivative of the function e^{x^{2}} is a common problem faced by high school and college students. Let's solve this step-by-step.

Solution

Let's start by making a substitution to simplify the problem.

Step 1: Substitution

We let e^{x^{2}} e^u. This substitution helps us to simplify the expression step-by-step.

Step 2: Taking the Natural Logarithm

Applying the natural logarithm on both sides, we get:

x^2 ln(e) u ln(e)

Since ln(e) 1, this simplifies to:

x^2 u

Step 3: Taking the Differential

Differentiating both sides with respect to x, we get:

2x dx du

Thus, we can express du/dx as:

du/dx 2x

Step 4: Applying Chain Rule

Now, using the chain rule to differentiate the original expression:

de^x^2/dx de^u/dx du/dx * e^u

Since du/dx 2x and e^u e^x^2, we can substitute these into the expression:

de^x^2/dx 2x * e^x^2

Thus, the final answer is:

2x * e^x^2

First Derivative of 1

Now, let's explore another common calculus problem: the first derivative of 1. This is a more straightforward example as we will see.

New Problem: First Derivative of e^x^2

All right, we have a similar problem in the form of the first derivative of e^{x^2} with respect to x.

Step 1: Equation Setup

Equation (1) states:

y e^{x^2} e^{2x}

Step 2: Differentiating y

Applying the chain rule for differentiation:

dy/dx 2e^(2x)

So, the first derivative of e^{x^2} with respect to x is:

2e^(2x)

Conclusion

Understanding these concepts is crucial for mastering calculus. The derivative of e^{x^2} and the derivative of e^{x^2} e^{2x} are important for broader applications in mathematics and physics. If you found this helpful, you might also be interested in exploring the following related topics:

The properties of exponential functions The chain rule in calculus Important derivatives in calculus

Feel free to dive into these and continue refining your calculus skills!