Differential coefficient of ₁0} w. r. t ₓ 10 is

Correct Answer is : - (log⁡x)2(log⁡10)2( x)²/( 10)²(log10)2(logx)2

Correct Answer is : - (log⁡x)2(log⁡10)2( x)²/( 10)²(log10)2(logx)2

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NIMCET 2018 Question Paper with solutions

Question : 20 of 120

Marks: +1, -0

\log_{10}

\log_x

10 is

Solution:

Let

u =

\log_{10}

x and v =

\log_x

u =

\frac{\log_e x}{\log_e 10}

, v =

\frac{\log_e 10}{\log_e x}

\frac{du}{dx}

\frac{1}{\log_e 10} \cdot \frac{1}{x}

[differentiating w.r.t x]

and

\frac{dv}{dx}

\log_e

10 .

\frac{1}{(\log_e x)^2} \cdot \frac{1}{x}

Now ,

\frac{du}{dv}

\frac{\frac{du}{dx}}{\frac{dv}{dx}}

\frac{ \frac{1}{\log_e 10} \cdot \frac{1}{x} }{ -\frac{\log_e 10}{(\log_e x)^2} \cdot \frac{1}{x} }

= -

\frac{1}{\log_e 10} \cdot \frac{1}{x} \cdot \frac{x}{\log_e 10} (\log_e x)^2

= -

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

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