5. continuity and differentiation — Maths STD 12 Science — Question

at $x=1$ we have

$1-2 \cot y-1=0$

$\Rightarrow$ cot $y=0 \quad \therefore \quad y=\pi / 2$

Differentiating $(i)$ w.r.t. $x,$ we have

$2{x^{2x}}(1 + \ln x) - 2\left[ {{x^x}\left( { - \cos e{c^2}y} \right)\frac{{dy}}{{dx}}} \right.$

$\left. { + \cot y \cdot {x^x}(1 + \ln x)} \right] = 0$

At $P(1, \pi / 2)$ we have

$2(1+\ln 1)-2\left[1(-1)\left(\frac{d y}{d x}\right)_{P}+0\right]=0$

$\Rightarrow 2+2\left(\frac{d y}{d x}\right)_{P}=0 \quad \therefore\left(\frac{d y}{d x}\right)_{P}=-1$