Problem 27 - Calculus Single Variable

Question

Calculate the derivative with respect to \(x\) of the given expression. \(\log _{10}(5 / x)\)

Step-by-Step Solution

Answer

The derivative of \( \log_{10}(5/x) \) with respect to \(x\) is \( -\frac{1}{x \ln(10)} \).

Step 1: Understand the Given Expression

The expression provided is \( \log_{10}(5/x) \). This is the logarithmic function with base 10 of the fraction \( \frac{5}{x} \). Our task is to find the derivative of this expression with respect to \(x\).

Step 2: Apply the Change of Base Formula

To simplify the expression, use the change of the base formula for logarithms: \( \log_{b}(a) = \frac{\ln(a)}{\ln(b)} \). This allows us to express the logarithm with base 10 in terms of the natural logarithm: \( \log_{10}(\frac{5}{x}) = \frac{\ln(\frac{5}{x})}{\ln(10)} \).

Step 3: Differentiate the Natural Logarithm

Differentiate \( \ln(\frac{5}{x}) \) with respect to \(x\). Using the chain rule, first differentiate the natural logarithm: \( \frac{d}{dx} \ln(\frac{5}{x}) = \frac{d}{dx}(\ln(5) - \ln(x)) \). This simplifies to \( 0 - \frac{1}{x} = -\frac{1}{x} \).

Step 4: Differentiate the Expression Using Constant Multiple Rule

The derivative of the expression \( \frac{1}{\ln(10)} \cdot \ln(\frac{5}{x}) \) can be found by multiplying the derivative of \( \ln(\frac{5}{x}) \) by the constant \( \frac{1}{\ln(10)} \). Thus, the derivative is \( \frac{1}{\ln(10)} \cdot (-\frac{1}{x}) \).

Step 5: Simplify the Derivative

Combine and simplify the expression: \( -\frac{1}{x \ln(10)} \). This is the derivative of the original expression with respect to \(x\).

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical functions that are essentially the inverse of exponential functions. In the context of calculus, the logarithmic function is often used as a tool for simplifying complex expressions before differentiation or integration. Each logarithmic function has a specific base, which means it shows how many times you need to multiply the base to get a certain number. For instance, in the expression \( \log_{10}(x) \), the base is 10.

• The logarithm with base 10 is known as the common logarithm.
• Logarithms can be used to solve exponential equations and are invaluable in negative exponential growth scenarios.
• In calculus, logarithms are helpful for taking derivatives of functions that involve multiplication or division of variables.

In our exercise, the given expression is \( \log_{10}(5/x) \). This indicates a logarithmic function with base 10 applied to the fraction \( \frac{5}{x} \). Understanding this setup is key to correctly applying the change of base formula and subsequently finding the derivative.

Chain Rule

The chain rule is a fundamental technique in calculus for differentiating composite functions. A composite function typically looks like \( f(g(x)) \), where you have an outer function \( f \) and an inner function \( g \).

• To differentiate such a function, apply the chain rule: \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \).
• You'll differentiate the outer function while keeping the inner function unchanged, then multiply by the derivative of the inner function.

In the solution to the exercise, we applied the chain rule to the natural logarithm \( \ln(\frac{5}{x}) \), noting that the inner function here is \( \frac{5}{x} \).
First, the derivative of the outer logarithmic function, \( \ln(u) \), is \( \frac{1}{u} \).
Then, consider the derivative of the inner function, which turns the expression into \( \ln(5) - \ln(x) \), simplifying with the chain rule to: \(-\frac{1}{x}\). This showcases the power of the chain rule for derivatives.

Change of Base Formula

The change of base formula is extremely useful when working with logarithms of different bases. It allows you to convert a logarithm from one base into another, typically into base \( e \) (natural logarithm) for easier differentiation.

• The formula is given by \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \).
• This formula is essential for simplifying logarithms into forms that are more manageable in calculus operations.

In our exercise, we had \( \log_{10}(\frac{5}{x}) \). By applying the change of base formula, this can be rewritten as \( \frac{\ln(\frac{5}{x})}{\ln(10)} \). This transformation allows us to manipulate \( \ln(\frac{5}{x}) \) directly when differentiating, significantly simplifying the calculus involved.