Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Differentiation

NDA 2018 · Mathematics · Differentiation

If y=| x|^ |x| , then what is the value of (d y)/(d x) at x=(π)/(6)...

If y=∣sin⁡x∣∣x∣y=|\sin x|^{|x|}, then what is the value of dydx\frac{d y}{d x} at x=π6x=\frac{\pi}{6} ?

Options

  1. A2−π6(6ln⁡2−3π)6\frac{2^{-\frac{\pi}{6}}(6 \ln 2-\sqrt{3} \pi)}{6}
  2. B2π6(6ln⁡2+3π)6\frac{2^{\frac{\pi}{6}}(6 \ln 2+\sqrt{3} \pi)}{6}
  3. C2−π6(6ln⁡2+3π)6\frac{2^{-\frac{\pi}{6}}(6 \ln 2+\sqrt{3} \pi)}{6}
  4. D2π6(6ln⁡2−3π)6\frac{2^{\frac{\pi}{6}}(6 \ln 2-\sqrt{3} \pi)}{6}

Correct answer

A. 2−π6(6ln⁡2−3π)6\frac{2^{-\frac{\pi}{6}}(6 \ln 2-\sqrt{3} \pi)}{6}

Step-by-step solution

y=(−sin⁡x)−xdydx=(−sin⁡)−x[xsin⁡x(−cos⁡x)+log⁡(−sin⁡x)⋅(−1)][dydx]x=−π6=(12)π/6[−π63−log⁡12]=2−π/6(6log⁡2−3π6)\begin{aligned} & y=(-\sin x)^{-x} \\ & \frac{d y}{d x}=(-\sin )^{-x}\left[\frac{x}{\sin x}(-\cos x)+\log (-\sin x) \cdot(-1)\right] \\ & {\left[\frac{d y}{d x}\right] x=-\frac{\pi}{6}=\left(\frac{1}{2}\right)^{\pi / 6}\left[-\frac{\pi}{6} \sqrt{3}-\log \frac{1}{2}\right]} \\ & =2^{-\pi / 6}\left(\frac{6 \log 2-\sqrt{3} \pi}{6}\right)\end{aligned}
Practice Differentiation on Quantrex Academy →

More from Differentiation

Passage: Let (e^y)^x-y=0 , where y is a function of x whose domain is (0,10] . Question… NDA 2026If [4] y =x+ x^2+4 , then what is x^2+4 \,(dy)/(dx) equal to ? NDA 2026Passage: Let (e^y)^x-y=0 , where y is a function of x whose domain is (0,10] . Question… NDA 2026For the following two (02) items: Consider the function f(x) = cases 4(5^x), & x What is… NDA 2025Consider the following for the two (02) items that follow: Let x = θ - θ and y = ^4θ… NDA 2025Consider the following for the two (02) items that follow: Let (x+y)^ p+q = x^p y^q… NDA 2025Consider the following for the two (02) items that follow: Let y = ^ -1 (x… NDA 2025For the following three (03) items: Let y = f(x) = x ^ -1 x 1 - x^2 + 1 - x^2 . If x = θ… NDA 2025Consider the following for the two (02) items that follow: Let (x+y)^ p+q = x^p y^q… NDA 2025For the following three (03) items: Let y = f(x) = x ^ -1 x 1 - x^2 + 1 - x^2 . What is… NDA 2025Consider the following for the two (02) items that follow: Let x = θ - θ and y = ^4θ… NDA 2025Paragraph: Let f:(-1,1) arrow R be a differentiable function with f(0)=-1 and f^ (0)=1 .… NDA 2024Let f(x) and g(x) be two functions such that g(x)=x-(1)/(x) and f o g(x)=x^3-(1)/(x^3) .… NDA 2024Paragraph: Let f:(-1,1) arrow R be a differentiable function with f(0)=-1 and f^ (0)=1 .… NDA 2024Let a function f be defined on ℝ-\ 0\ and 2 f(x)+f((1)/(x))=x+3 If f is differentiable… NDA 2023If f(x)=| | x| | where 0 < x < 1 , then what is f^ (0.5) equal to? NDA 2023 Full Differentiation list Differentiation JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs