Question

найти d^2y/dх^2 функции, заданной параметрически:
Cистема:
y=arcctgt
y=ln(2+t^2)

Answer 1

$left { {{y=arcctgt} atop {x=ln(2+t^2)}} right. ; ; ; ; ; ; ; y'_{x}= frac{y'_{t}}{x'_{t}} \\y'_{t}=-frac{1}{1+t^2}\\x'_{t}=frac{2t}{2+t^2}\\y'_{x}=-frac{2+t^2}{(1+t^2)cdot 2t}\\y''_{xx}= frac{(y'_{x})'_{t}}{x'_{t}} \\(y'_{x})'_{t}=(-frac{2+t^2}{2t^3+2t})'_{t}=- frac{2t(2t^3+2t)-(2+t^2)(6t^2+2)}{(2t^3+2t)^2} =\\=- frac{4t^4+4t^2-12t^2-4-6t^4-2t^2}{4t^2cdot (1+t^2)^2}= -frac{-2t^4-10t^2-4}{4t^2(1+t^2)^2} = frac{t^4+5t^2+2}{t^2(1+t^2)^2}$

$y''_{xx}= frac{t^4+5t^2+2}{t^2(1+t^2)^2} : frac{2t}{2+t^2} = frac{(2+t^2)(t^4+5t^2+2)}{2t^3(1+t^2)^2}$