Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Basics of Mathematics

NDA 2016 · Mathematics · Basics of Mathematics

If y= _ 10 x+ _ x 10+ _ x x+ _ 10 10 then what is ( dy dx )_ x=10...

If y=log⁡10x+log⁡x10+log⁡xx+log⁡1010\mathrm{y}=\log _{10} \mathrm{x}+\log _{\mathrm{x}} 10+\log _{\mathrm{x}} \mathrm{x}+\log _{10} 10 then what is (dydx)x=10\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\mathrm{x}=10} equal to?

Options

  1. A10
  2. B2
  3. C1
  4. D0

Correct answer

D. 0

Step-by-step solution

y=log⁡10x+log⁡x10+log⁡xx+log⁡1010y=log⁡10x+log⁡x10+1+1\begin{aligned} & y=\log _{10} x+\log _x 10+\log _x x+\log _{10} 10 \\ & y=\log _{10} x+\log _x 10+1+1 \end{aligned} Differentiating equation w.r.t. x dydx=1xlog⁡e10−1(log⁡10x)2⋅1(xlog⁡10)=1xlog⁡e10[1−1(log⁡10x)2](dydx)x=10=110log⁡e10[1−1]=0[ Note: log⁡x10=log⁡1010log⁡10x=1log⁡10xddx[1log⁡10x]=−(log⁡10x)−2×1xlog⁡e10=−1(log⁡10x)2xlog⁡e10]\begin{aligned} & \frac{d y}{d x}=\frac{1}{x \log _e 10}-\frac{1}{\left(\log _{10} x\right)^2} \cdot \frac{1}{(x \log 10)} \\ & =\frac{1}{x \log _e 10}\left[1-\frac{1}{\left(\log _{10} x\right)^2}\right] \\ & \left(\frac{d y}{d x}\right)_{x=10}=\frac{1}{10 \log _e 10}[1-1]=0 \\ & {\left[\begin{array}{rl} \text{ Note: } \log _x 10 & =\frac{\log _{10} 10}{\log _{10} x}=\frac{1}{\log _{10} x} \\ \frac{d}{d x}\left[\frac{1}{\log _{10} x}\right] & =-\left(\log _{10} x\right)^{-2} \times \frac{1}{x \log _e 10} \\ & =-\frac{1}{\left(\log _{10} x\right)^2 x \log _e 10} \end{array}\right]} \end{aligned}
Practice Basics of Mathematics on Quantrex Academy →

More from Basics of Mathematics

All Basics of Mathematics questions Full Basics of Mathematics list Basics of Mathematics JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs