Let y be an implicit function of x defined by x^2 x-2 x^x cot y-1=0. Then y^prime(1) equals

$x^{2 x}-2 x^x \cot y-1=0$x2x−2xxcoty−1=0 Now $x=1$x=1, $$1-2 \cot y-1=0 \Rightarrow \cot y=0 \Rightarrow y=\frac{\pi}{2}$$1−2coty−1=0⇒coty=0⇒y=2π​ Now differentiating eq. (1) w.r.t. ' $x$x ' $$2 x^{2 x}(1+\log x)-2\left[x^x\left(-\operatorname{cosec}^2 y\right) \frac{d y}{d x}+\cot y x^x(1+\log x)\right]=0$$2x2x(1+logx)−2[xx(−cosec2y)dxdy​+cotyxx(1+logx)]=0 Now at $\left(1, \frac{\pi}{2}\right)$(1,2π​) $$\begin{aligned} & 2(1+\log 1)-2\left(1(-1)\left(\frac{d y}{d x}\right)_{\left(1, \frac{\pi}{2}\right)}+0\right)=0 \\ & \Rightarrow 2+2\left(\frac{d y}{d x}\right)_{\left(1, \frac{\pi}{2}\right)}=0 \Rightarrow\left(\frac{d y}{d x}\right)_{\left(1, \frac{\pi}{2}\right)}=-1 \end{aligned}$$​2(1+log1)−2(1(−1)(dxdy​)(1,2π​)​+0)=0⇒2+2(dxdy​)(1,2π​)​=0⇒(dxdy​)(1,2π​)​=−1​