Differentiation Question 51
Question: Let y be an implicit function of x defined by $ x^{2x}-2x^{x}\cot y-1=0 $ . Then y’(1) equals
Options:
A) -1
B) 1
C) $ log2 $
D) $ -log2 $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ x^{2x}-2x^{x}\cot y-1=0 $ (i) Now at x=1, $ 1-2\cot y-1=0\Rightarrow \cot y=0\Rightarrow y=\frac{\pi }{2} $
Now differentiating (i) w.r.t,x, we get $ 2x^{2x}(1+logx)-2[ x^{x}(-cosec^{2}y)\frac{dy}{dx}+\cot yx^{x}(1+logx) ]=0 $ Now at $ (1,\pi /2) $ , $ 2(1+log1)-2[ 1(-1){{( \frac{dy}{dx} )} _{1,\pi /2}}+0 ]=0 $
$ \Rightarrow 2+2{{( \frac{dy}{dx} )} _{(1,\pi /2)}}=0 $
$ \Rightarrow {{( \frac{dy}{dx} )} _{(1,\pi /2)}}=-1 $
BETA
