Differentiating log(x)/x^2: A Comprehensive Guide

In this comprehensive guide, we will explore the differentiation of the function (frac{log x}{x^2}) using the Quotient Rule. We will break down the process into clear, detailed steps, making it easier to understand even for those who are new to calculus.

Understanding the Functions

It is important to clarify that in this context, we are dealing with the logarithm with base 10, represented as (log x). This means (log x log_{10}x).

The Quotient Rule

The Quotient Rule is used to differentiate functions that are quotients of two functions. It states that:

[left( frac{u}{v} right)' frac{v left(frac{du}{dx}right) - u left(frac{dv}{dx}right)}{v^2}]

Application of the Quotient Rule

Let's apply the Quotient Rule to differentiate (frac{log x}{x^2}).

Set (u log x) and (v x^2).

Compute (frac{du}{dx}):

Set (log x y). Then, (10^y x). Take the natural logarithm of both sides: (y ln 10 ln x). Take the derivative of both sides: (ln 10 frac{dy}{dx} frac{1}{x}). Solve for (frac{dy}{dx}): (frac{d}{dx} log x frac{1}{x ln 10}).

Compute (frac{dv}{dx}) using the Power Rule: (frac{d}{dx} x^2 2x).

Substitute (u), (v), (frac{du}{dx}), and (frac{dv}{dx}) into the Quotient Rule:

[frac{x^2 left(frac{1}{x ln 10} right) - log x (2x)}{x^4}]

Simplify the expression:

[frac{x - 2x log x}{x^4 ln 10}]

Further simplify the expression by multiplying the numerator and denominator by (ln 10):

[frac{x ln 10 - 2x^2 ln 10 log x}{x^4 (ln 10)^2}]

Factor out common terms:

[frac{x - 2x ln x}{x^4 ln 10} frac{1 - 2 ln x}{x^3 ln 10}]

Convert (ln x) to base (e) for clarity (if needed):

[frac{1 - 2 ln x}{x^3 ln 10} frac{ln e - 2 ln x}{x^3}]

Alternative Methods

Direct Quotient Rule Application:

[frac{d}{dx} frac{log x}{x^2} frac{x^2 cdot frac{1}{x ln 10} - log x cdot 2x}{x^4} frac{1 - 2 log x}{x^3 ln 10}]

Using the log of the product:

[log (log x) log x - 2 log x] [frac{1}{log x} cdot frac{d}{dx} log x frac{frac{1}{x} - 2 log x}{x log x}] [frac{d}{dx} frac{log x}{x^2} frac{1 - 2 log x}{x^3}]

Logarithmic Differentiation:

[log left( frac{log x}{x^2} right) log (log x) - 2 log x] [frac{1}{frac{log x}{x^2}} cdot frac{d}{dx} frac{log x}{x^2} frac{1 - 2 log x}{x^2}] [frac{d}{dx} frac{log x}{x^2} frac{1 - 2 log x}{x^3}]

Conclusion

Differentiating the function (frac{log x}{x^2}) can be approached in several ways, each leading to the same result. By using the Quotient Rule, logarithmic properties, and even logarithmic differentiation, we can simplify and clarify the process of differentiation in this context.

Understanding these methods not only helps in solving similar problems but also deepens one’s understanding of calculus and the properties of logarithmic functions. Whether you are an advanced calculus student or a professional, these techniques are valuable tools in your mathematical toolkit.

Stay tuned for more detailed guides and solutions to other mathematical challenges!