The problem asks to find the differential coefficient (derivative) of log10 x with respect to logx 10.

Logarithm Differentiation Setup

To find the derivative of one logarithmic function with respect to another, we first define:

• y = log10 x
• z = logx 10

We need to calculate dy/dz.

Key Calculus Principles

This problem involves standard calculus rules:

• Change of Base Formula: Converts logarithms to a common base. loga b = ln(b) / ln(a).
• Derivative of ln(x): d/dx (ln x) = 1/x.
• Chain Rule: Used to find derivatives involving composite functions. dy/dz = (dy/dx) / (dz/dx).

Step-by-Step Derivative Calculation

1. Express y and z using natural logarithms (ln):

   • y = log10 x = \(\dfrac{\ln x}{\ln 10}\)
   • z = logx 10 = \(\dfrac{\ln 10}{\ln x}\)

2. Find the derivative of y with respect to x (dy/dx):

   dy/dx = \(\dfrac{d}{dx}\left(\dfrac{\ln x}{\ln 10}\right)\)

   Since ln(10) is a constant:

   dy/dx = \(\dfrac{1}{\ln 10} \cdot \dfrac{1}{x} = \dfrac{1}{x \ln 10}\)

3. Find the derivative of z with respect to x (dz/dx):

   Rewrite z as z = ln(10) * (ln x)-1.

   dz/dx = \(\dfrac{d}{dx}\left(\ln 10 \cdot (\ln x)^{-1}\right)\)

   Using the power rule and chain rule:

   dz/dx = \(\ln 10 \cdot (-1) (\ln x)^{-2} \cdot \dfrac{1}{x} = -\dfrac{\ln 10}{x (\ln x)^2}\)

4. Calculate dy/dz:

   dy/dz = \(\dfrac{dy/dx}{dz/dx} = \dfrac{1 / (x \ln 10)}{-\ln 10 / (x (\ln x)^2)}\)

   Simplify the expression:

   dy/dz = \(\dfrac{1}{x \ln 10} \cdot \left(-\dfrac{x (\ln x)^2}{\ln 10}\right) = -\dfrac{(\ln x)^2}{(\ln 10)^2}\)

5. Express the result using base-10 logarithms:

   Using log10 x = ln(x) / ln(10):

   dy/dz = \(= -\left(\dfrac{\ln x}{\ln 10}\right)^2 = -(\log_{10} x)^2\)

Comparing with Provided Options

We now compare our result - (log10 x)2 with the options. Conventionally, log x implies log10 x and log 10 implies log10 10 = 1.

• Option 1: \(-\dfrac{(\log x)^2}{(\log 10)^2}\) evaluates to \(-(\log_{10} x)^2 / (1)^2 = -(\log_{10} x)^2\). This matches our result.
• Option 2: \(\dfrac{(\log_{10}x)^2}{(\log 10)^2}\) evaluates to \((log_{10} x)^2\). Incorrect.
• Option 3: \(\dfrac{(\log_x 10)^2}{(\log 10)^2}\) evaluates to \((log_x 10)^2\). Incorrect.
• Option 4: \(-\dfrac{(\log 10)^2}{(\log x)^2}\) evaluates to \(- (1)^2 / (\log_{10} x)^2 = -1 / (\log_{10} x)^2\). Incorrect.

Final Answer Derivation

The derived result matches Option 1. The differential coefficient of log10 x with respect to logx 10 is \(-\dfrac{(\log x)^2}{(\log 10)^2}\).