Question

The derivative of \(log_{10}x\) with respect to \(log_x10\) is

• A. \(-log_{10}x^2\)
• B. \(-(log_{10}x)^2\)
• C. \((log_x10)^2\)
• D. \(\frac{x^2}{100}\)

Hint:

Write both functions in natural logarithms and divide the derivatives.

Answer 1

The Correct Option is B

Step 1: Write in Natural Logs:
Let \(u=\log_{10}x=\dfrac{\ln x}{\ln10}\) and \(v=\log_x10=\dfrac{\ln10}{\ln x}\).

Step 2: Differentiate Both:
\(\dfrac{du}{dx}=\dfrac{1}{x\ln10}\). \(\dfrac{dv}{dx}=-\dfrac{\ln10}{x(\ln x)^2}\).

Step 3: Divide:
\[ \frac{du}{dv}=\frac{1/(x\ln10)}{-\ln10/(x(\ln x)^2)}=-\frac{(\ln x)^2}{(\ln10)^2}=-\left(\log_{10}x\right)^2 \]

Step 4: Check the Options:
Option (C) has the wrong sign. Options (A) and (D) do not match the form \((\ln x/\ln10)^2\). So (B) is correct.

Final Answer:
The derivative is \(-(\log_{10}x)^2\), option (B). \[ \boxed{\text{(B) } -(\log_{10}x)^2} \]