Exploring the Derivative of a Complex Function: y x^x ln x

Understanding the derivative of complex functions is crucial in many areas of advanced mathematics and its applications. In this article, we will delve into the intricacies of finding the derivative of a particular complex function, namely y x^x ln x. This function introduces challenges related to both exponential and logarithmic terms, making the process of differentiation non-trivial and fascinating. We will employ the method of logarithmic differentiation to derive the derivative in a step-by-step manner.

Introduction to Logarithmic Differentiation

Logarithmic differentiation is a powerful technique for finding the derivatives of complex functions. Essentially, if we have a function in the form of y f(x)^{g(x)}, we can take the natural logarithm of both sides and then differentiate both sides with respect to x. This initial step simplifies the differentiation process, making it more manageable.

Applying Logarithmic Differentiation

Let's start by applying logarithmic differentiation to our function y x^x ln x.

Step 1: Taking the Natural Logarithm

We begin by taking the natural logarithm (ln) of both sides:

ln y ln (x^x ln x)

Using the properties of logarithms, we can break this down as:

ln y x ln (x ln x)

Applying the logarithm product rule:

ln y x [ln (x ln x)]

Step 2: Differentiating Both Sides

To find the derivative, we differentiate both sides with respect to x. First, differentiate the left side:

(1/y) dy/dx

For the right side, we use the product rule on the term x [ln (x ln x)]:

d/dx [x ln (x ln x)] ln (x ln x) x ln (1 1/lx)

Combining these results, we have:

(1/y) dy/dx ln (x ln x) x ln (1 1/lx)

Simplifying the right side:

(1/y) dy/dx ln (x ln x) x [ln (x 1/lx)]

Step 3: Solving for dy/dx

To isolate dy/dx, we multiply both sides by y:

dy/dx y [ln (x ln x) x [ln (x 1/lx)]]

Substitute the original function y x^x ln x back into the expression:

dy/dx x^x ln x [ln (x ln x) x [ln (x 1/lx)]]

Conclusion

In this article, we explored the derivative of the complex function y x^x ln x using logarithmic differentiation. This method is particularly powerful for functions involving both exponential and logarithmic terms. The step-by-step process shown here helps to break down the problem and make it more tractable. Understanding such derivatives is essential in fields such as calculus, physics, and engineering.

Key Points:

Logarithmic differentiation simplifies complex functions. The derivative of y x^x ln x is dy/dx x^x ln x [ln (x ln x) x [ln (x 1/lx)]]. Applying the product rule and product-to-sum identities is crucial in solving complex derivatives.