Understanding the Derivative of 1/x logx

In this article, we will explore the differentiation of the function 1/x logx. This is a common problem encountered in calculus and understanding it thoroughly is crucial for advanced mathematical analysis and applications. The solution to this problem involves the product rule and the chain rule.

Introduction to the Problem

We are given the function f(x) 1/x log x, which can be rewritten as f(x) log x * x^(-1). To find the derivative of this function, we will follow a step-by-step approach, breaking it down into simpler parts and applying the appropriate rules.

Step-by-Step Derivation

To find the derivative of f(x) log x * x^(-1), let's use the product rule. The product rule states that if f(x) g(x) * h(x), then the derivative f'(x) g'(x) * h(x) g(x) * h'(x).

Identifying g(x) and h(x)

Let us define g(x) log x and h(x) x^(-1).

Finding g'(x) and h'(x)

The derivative of g(x) log x is g'(x) 1/x.

The derivative of h(x) x^(-1) is h'(x) -x^(-2) or -1/x^2.

Applying the Product Rule

Using the product rule, we get:

f'(x) g'(x) * h(x) g(x) * h'(x)

f'(x) (1/x) * x^(-1) log x * (-1/x^2)

Simplifying the expression, we get:

f'(x) 1/x^2 - log x / x^2

Factoring out 1/x^2, we obtain:

f'(x) (1 - log x) / x^2

Therefore, the derivative of 1/x log x is:

boxed{(1 - log x) / x^2}

Alternative Methods and Considerations

Here, we also discuss the alternative ways to derive the same result and considerations for different bases of the logarithm.

Using the Quotient Rule

Alternatively, we can approach this using the quotient rule. The quotient rule states that if f(x) u(x) / v(x), then f'(x) (u'v - uv') / v^2.

Here, let f(x) 1/x log x, which can be written as f(x) u(x) / v(x) with u(x) log x and v(x) x.

The derivatives are u'(x) 1/x and v'(x) 1. Applying the quotient rule:

f'(x) (u'v - uv') / v^2 (1/x * x - log x * 1) / x^2

Simplifying, we get:

f'(x) (1 - log x) / x^2

Logarithmic Base Considerations

It's important to note that if the base of the logarithm log x is not specified, it is often assumed to be the natural logarithm ln x. However, if it is a different base, say base b, then the logarithm should be converted accordingly.

Conclusion

In conclusion, the derivative of the function 1/x log x is boxed{(1 - log x) / x^2}. This result has important applications in calculus and analysis, particularly when dealing with logarithmic functions.

For further exploration and understanding, consider practicing similar problems and reviewing the concepts of product rule, quotient rule, and logarithmic differentiation.