NDA 2022
Exam NDA
Year 2022
Subject Mathematics
Chapter Differentiation NDA 2022 · Mathematics · Differentiation
If x^y y^x=1 , then what is (d y)/(d x) at (1,1) equal to?
If
x y y x = 1 x^y y^x=1 x y y x = 1 , then what is
d y d x \frac{d y}{d x} d x d y at
( 1 , 1 ) (1,1) ( 1 , 1 ) equal to?
Step-by-step solution
\mathrmx y \mathrmy x = 1 \mathrmx^{\mathrm{y}} \mathrmy^{\mathrm{x}}=1 \mathrmx y \mathrmy x = 1
Taking log both sides
y log x + x log y = 0 y ′ log x + y x + log y + x y y ′ = 0 \begin{aligned} & y \log x+x \log y=0 \\ & y^{\prime} \log x+\frac{y}{x}+\log y+\frac{x}{y} y^{\prime}=0\end{aligned} y log x + x log y = 0 y ′ log x + x y + log y + y x y ′ = 0
put
x = 1 , y = 1 \mathrm{x}=1, \mathrm{y}=1 x = 1 , y = 1 \mathrmy ′ log ( 1 ) + 1 + log ( 1 ) + \mathrmy ′ = 0 0 + 1 + 0 + \mathrmy ′ = 0 ⇒ y 1 = − 1 \begin{aligned} & \mathrmy^{\prime} \log (1)+1+\log (1)+\mathrmy^{\prime}=0 \\ & 0+1+0+\mathrmy^{\prime}=0 \Rightarrow \mathrm{y}^1=-1\end{aligned} \mathrmy ′ log ( 1 ) + 1 + log ( 1 ) + \mathrmy ′ = 0 0 + 1 + 0 + \mathrmy ′ = 0 ⇒ y 1 = − 1 $$
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