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: Use the Reciprocal Relation:
$v=1/u$ since $\log_x10=1/\log_{10}x$.

Step 2: Differentiate v With Respect to u:
$\dfrac{dv}{du}=-\dfrac1{u^2}$, so $\dfrac{du}{dv}=-u^2$.

Step 3: Answer:
$u=\log_{10}x$, so the result is $-(\log_{10}x)^2$. Option (B).

Final Answer:
Option (B). \[ \boxed{\text{(B) } -(\log_{10}x)^2} \]