Differential co-efficient of $\log_{10}x

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Continuity and Differentiability

Question:

Differential co-efficient of $\log_{10}x$ w.r.t. $\log_{x} 10$ is

Options:

$-\frac{(\log x)^2}{(\log 10)^2}$

$\frac{(\log_{10} x)^2}{(\log 10)^2}$

$\frac{(\log_{x} 10)^2}{(\log 10)^2}$

$-\frac{(\log 10)^2}{(\log x)^2}$

Correct Answer:

$-\frac{(\log x)^2}{(\log 10)^2}$

Explanation:

Let $y =\log_{10}x$ and $z=\log_x 10=\frac{1}{\log_{10}x}=\frac{1}{y}$

$∴y=\frac{1}{z}∴\frac{dy}{dz}=-\frac{1}{z^2}=-\frac{1}{(\log_x10)^2}=-\frac{(\log x)^2}{(\log 10)^2}$