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UPSC NDA I 2016 Mathematics Solved Paper

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Question : 119 of 120

Marks: +1, -0

If

y = \log_{10} x + \log_{x} 10 + \log_{x} x + \log_{10} 10

\left( \frac{dy}{dx} \right)_{x=10}

equal to?

10
2
1
0

Solution:

Concept:

Use logarithmic properties

\log_a a = 1

and

\log_a b = \frac{1}{\log_b a}

to simplify, then differentiate.

Explanation:

Given

y = \log_{10} x + \log_x 10 + \log_x x + \log_{10} 10

.

Simplify using

\log_a a = 1

:

\log_x x = 1

,

\log_{10} 10 = 1

.

So

y = \log_{10} x + \log_x 10 + 2

.

Convert to natural logs:

\log_{10} x = \frac{\ln x}{\ln 10}

,

\log_x 10 = \frac{\ln 10}{\ln x}

.

Thus

y = \frac{\ln x}{\ln 10} + \frac{\ln 10}{\ln x} + 2

.

Differentiate with respect to

x

:

\frac{dy}{dx} = \frac{1}{\ln 10} \cdot \frac{1}{x} + \ln 10 \cdot \left( -\frac{1}{(\ln x)^2} \cdot \frac{1}{x} \right)

.

So

\frac{dy}{dx} = \frac{1}{x\ln 10} - \frac{\ln 10}{x(\ln x)^2}

.

Evaluate at

x=10

:

\ln x = \ln 10

.

Then

\frac{dy}{dx}\big|_{x=10} = \frac{1}{10\ln 10} - \frac{\ln 10}{10(\ln 10)^2} = \frac{1}{10\ln 10} - \frac{1}{10\ln 10} = 0

.

Answer:

0

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