Understanding the Partial Derivative of the Product Rule: x log x

Introduction to the Concept

In calculus, the concept of a derivative is fundamental in understanding the rate of change of a function with respect to its variables. When dealing with a single variable function, the chain and product rules are essential tools. This article aims to explore the partial derivative of the product rule in a specific function, f(x) x · log(x), focusing on the step-by-step differentiation using the product rule.

Derivative vs Partial Derivative

To begin, it is important to clarify the distinction between a derivative and a partial derivative. In the context of the function f(x) x · log(x), despite the name, we are only dealing with one variable, x. Consequently, the term "partial derivative" might seem redundant in this scenario. However, for the sake of consistency, we will handle this function as a single-variable derivative.

Basic Definition

The function in question is:

f(x) x · log(x)

Applying the Product Rule

For the sake of this discussion, let's assume we are differentiating u(x) · v(x), where u(x) x and v(x) log(x). The product rule of differentiation states:

frac{d}{dx}[u(x) · v(x)] u'(x) · v(x) u(x) · v'(x)

Step 1: Differentiate u(x) and v(x)

Let's start by finding the derivatives of u(x) and v(x) first:

u(x) x u'(x) 1 v(x) log(x) v'(x) frac{1}{x} (Using the derivative formula for logarithmic functions, frac{d}{dx}[log(x)] frac{1}{x})

Step 2: Apply the Product Rule

Now, we proceed to apply the product rule using the derivatives:

frac{d}{dx}[x · log(x)] 1 · log(x) x · frac{1}{x}

frac{d}{dx}[x · log(x)] log(x) 1

Conclusion

Hence, the derivative of the function f(x) x · log(x) is:

frac{d}{dx}[x · log(x)] 1 log(x)

Summary of Key Points

The function in focus is a logarithmic function multiplied by a linear function: x · log(x). The key concept here is the product rule of differentiation, which simplifies the process of finding the derivative of a product of functions. The final derivative is log(x) 1, indicating the rate of change of the function with respect to x.

Further Exploration

This understanding provides a foundation for more complex problems involving multiple variables and different functions. Practitioners of calculus and related fields can utilize this knowledge to explore advanced topics in mathematical analysis and applications in engineering, physics, and economics.