Question

50 баллов помогите пожалуйста

Answer 1

$1); ; frac{y}{x}=arctg(xy)\\frac{y'x-x'y}{x^2}=frac{1}{1+(xy)^2}cdot (x'y+xy'); ; ; ; ; [; x'=1; ]\\frac{y'x-y}{x^2}=frac{y+xy'}{1+x^2y^2}\\frac{y'}{x}-frac{y}{x^2}=frac{y}{1+x^2y^2}+frac{xy'}{1+x^2y^2}\\y'cdot Big (frac{1}{x}-frac{x}{1+x^2y^2}Big )=frac{y}{1+x^2y^2}+frac{y}{x^2}\\y'=frac{x^2y+y+x^2y^3}{x^2(1+x^2y^2)}:frac{1+x^2y^2-x^2}{x(1+x^2y^2)}\\y'=frac{x^2y+y+x^2y^3}{x(1+x^2y^2-x^2)}\\y'=frac{y(x^2+1+x^2y^2)}{x(-x^2+1+x^2y^2)}$

$2); ; x-3y+e^{y}-5=0\\x'-3y'+e^{y}cdot y'-0=0\\y'cdot (e^{y}-3)=-1\\y'=-frac{1}{e^{y}-3}\\y'=frac{1}{3-e^{y}}\\3); ; left { {{x=lnfrac{t^2-1}{4}} atop {y=sint}} right.\\x'_{t}=frac{4}{t^2-1}cdot frac{2t}{4}=frac{2t}{t^2-1}\\y'_{t}=cost\\y'_{x}=frac{y'_{t}}{x'_{t}}=frac{(t^2-1)cdot cost}{2t}$