Question

Помогите найти первую и вторую производную

Answer 1

$left { {{x=sqrt{t^2-1}} atop {y=frac{t+1}{sqrt{t^2-1}}}} right. \\x'_{t}=frac{2t}{2sqrt{t^2-1}}; ;\\y'_{t}=frac{sqrt{t^2-1}-(t+1)cdot frac{t}{sqrt{t^2-1}}}{t^2-1}= frac{t^2-1-t(t+1)}{(t^2-1)^{frac{3}{2}}} = frac{-t-1}{(t^2-1)^{3/2}} ; ;\\y'_{x}= frac{y'_{t}}{x'_{t}}= frac{-(t+1)sqrt{t^2-1}}{tcdot (t^2-1)^{3/2}}=-frac{t+1}{tcdot (t^2-1)}=- frac{t+1}{t^3-t} ; ;\\(y'_{x})'_{t}=- frac{t^3-t-(t+1)(3t^2-1)}{(t^3-t)^2} = -frac{t^3-t-3t^3+t-3t^2+1}{t^2(t^2-1)^2} =$

$=frac{2t^3+3t^2-1}{t^2(t^2-1)^2}; ;\\y''_{xx}= frac{(y'_{x})'_{t}}{x'_{t}}=frac{(2t^3+3t^2-1)sqrt{t^2-1}}{t^3(t^2-1)^2} =frac{2t^3+3t^2-1}{t^3(t^2-1)^{3/2}}; ;$