Question

Помогите, пожалуйста, найти производную у' по х

Answer 1

$17); ; {; x=lnfrac{1}{sqrt{1-t^4}} ; ,; ; y=arcsinfrac{1-t^2}{1+t^2}; }\\x'_{t}=sqrt{1-t^4}cdot frac{-frac{-4t^3}{2sqrt{1-t^4}}}{1-t^4}=frac{2t^3}{1-t^4}; ,\\y'_{t}=frac{1}{sqrt{1-(frac{1-t^2}{1+t^2})^2}}cdot frac{-2t(1+t^2)-2t(1-t^2)}{(1+t^2)^2}=frac{1}{sqrt{frac{1+2t^2+t^4-1+2t^2-t^4}{(1+t^2)^2}}}cdot frac{-2t-2t^3-2t+2t^3}{(1+t^2)^2}=\\=frac{1+t^2}{sqrt{4t^2}}cdot frac{-4t}{(1+t^2)^2}=-frac{2}{1+t^2}\\y'_{x}=frac{y'_{t}}{x'_{t}}=-frac{2}{1+t^2}cdot frac{1-t^4}{2t^3}=-frac{(1-t^2)(1+t^2)}{(1+t^2)cdot t^3}=-frac{1-t^2}{t^3}$

$2); ; {; x=ln(ctgt); ; ,; ; y=frac{1}{cos^2t}; }\\x'_{t}=frac{1}{ctgt}cdot frac{-1}{sin^2t}=-frac{sint}{costcdot sin^2t}=-frac{1}{sintcdot cost}=-frac{2}{sin2t}\\y'_{t}=frac{-2costcdot (-sint)}{cos^4t}=frac{sin2t}{cos^4t}\\y'_{x}=frac{sin2t}{cos^4t}cdot frac{sin2t}{-2}=-frac{sin^22t}{2cos^4t}$

$3); ; {x=ln(t+sqrt{t^2+1)}; ; ,; ; y=tsqrt{t^2+1}; }\\x'_{t}=frac{1}{t+sqrt{t^2+1}}cdot (1+frac{2t}{2sqrt{t^2+1}})=frac{1}{t+sqrt{t^2+1}}cdot frac{sqrt{t^2+1}+t}{sqrt{t^2+1}}=frac{1}{sqrt{t^2+1}}\\y'_{t}=sqrt{t^2+1}+tcdot frac{2t}{2sqrt{t^2+1}}=frac{t^2+1+2t^2}{sqrt{t^2+1}}=frac{3t^2+1}{sqrt{t^2+1}}\\y'_{x}=frac{3t^2+1}{sqrt{t^2+1}}cdot frac{sqrt{t^2+1}}{1}=3t^2+1$