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Mathematics Question on limits and derivatives

Differential co-efficient of log⁡10x\log_{10} x w.r.t. logx10log_x 10 is

A

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

B

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

C

(log⁡x10)2(log⁡10)2 \frac{(\log_x 10) 2}{(\log 10)^2}

D

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

Answer

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

Explanation

Solution

Let y=log⁡10x and z=log⁡x10=1log⁡10x=1yy = \log_{10} x \,and\,z = \log_{x} 10 = \frac{1}{\log_{10} x} = \frac{1}{y} ∴y=1z∴dydz=−1z2.\therefore y = \frac{1}{z} \therefore \frac{dy}{dz} = - \frac{1}{z^{2}} . =−1(log⁡x10)2=−(log⁡x)2(log⁡10)2= - \frac{1}{\left(\log_{x} 10\right)^{2}} = - \frac{\left(\log x\right)^{2}}{\left(\log10\right)^{2}}