Mathematics - Differentiation Question with Solution | TestHub

$x^{2x}-2x^x\cot y-1=0$ Now $x=1$, $$1-2\cot y-1=0\Rightarrow \cot y=0\Rightarrow y=\frac{\pi }{2}$$ Now differentiating eq. (1) w.r.t. ' $x$ ' $$2x^{2x}(1+\log x)-2\left[x^x\left(-{\mathrm{cosec}}^{2}y\right)\frac{dy}{dx}+\cot y{x}^x(1+\log x)\right]=0$$ Now at $\left(1,\frac{\pi }{2}\right)$ $$\begin{array}{rl}& 2(1+\log 1)-2\left(1(-1){\left(\frac{dy}{dx}\right)}_{\left(1,\frac{\pi }{2}\right)}+0\right)=0\\ & \Rightarrow 2+2{\left(\frac{dy}{dx}\right)}_{\left(1,\frac{\pi }{2}\right)}=0\Rightarrow {\left(\frac{dy}{dx}\right)}_{\left(1,\frac{\pi }{2}\right)}=-1\end{array}$$