Question

Найти производную функции
1) x^2/x^3+1
2)lnx/1-x

Answer 1

$1)\; \; y=\frac{x^2}{x^3+1}\\\\y'=\frac{2x\cdot (x^3+1)-x^2\cdot 3x^2}{(x^3+1)^2}=\frac{-x^4+2x}{(x^3+1)^2}=\frac{x\, (2-x^3)}{(x^3+1)^2}\\\\\\2)\; \; y=\frac{lnx}{1-x}\\\\y'=\frac{\frac{1}{x}\cdot (1-x)-lnx\cdot (-1)}{(1-x)^2}=\frac{\frac{1}{x}-1+lnx}{(1-x)^2}=\frac{1-x+x\cdot lnx}{x\, (1-x)^2}$