Question

Знайдіть похідну $\frac{d^{2} y}{dx^{2} }$ функції, заданої параметрично:

$x = cos(2t) - ln(ctg(t)), y = sin(2t)$

Answer 1

$\left\{\begin{array}{l}x=cos2t-ln(ctgt)\\y=sin2t\end{array}\right\\\\\\y'_{t}=2cos2t\ \ ,\\\\x'_{t}=-2sin2t-\dfrac{1}{ctgt}\cdot \dfrac{-1}{sin^2t}=-2sin2t+\dfrac{1}{cost\cdot sint}=-2sin2t+\dfrac{2}{sin2t}=\\\\\\=\dfrac{2-2sin^22t}{sin2t}=\dfrac{2(1-sin^22t)}{sin2t}=\dfrac{2cos^22t}{sin2t}\\\\\\y'_{x}=\dfrac{y'_{t}}{x'_{t}}=\dfrac{2cos2t\cdot sin2t}{2cos^22t}=\dfrac{sin4t}{2\, cos^22t}$

$y''_{tt}=(2cos2t)'_{t}=-4sin2t\\\\\\y''_{xx}=\dfrac{y''_{tt}}{x'_{t}}=\dfrac{-4sin2t\cdot sin2t}{2cos^22t}=-\dfrac{2sin^22t}{cos^22t}=-2\, tg^22t$