Tính đạo hàm của hàm số sau \[y = \frac{{2x + 1}}{{x + 2}}\]

Bước 1:

\[y' = {\left( {3x - 1} \right)^\prime }.\sqrt {{x^2} + 1} + \left( {3x - 1} \right).{\left( {\sqrt {{x^2} + 1} } \right)^\prime }\]

Bước 2:

\[\begin{array}{*{20}{l}}{ = 3.\sqrt {{x^2} + 1} + \left( {3x - 1} \right).\frac{{{{\left( {{x^2} + 1} \right)}^\prime }}}{{2.\sqrt {{x^2} + 1} }}}\\{ = 3.\sqrt {{x^2} + 1} + \left( {3x - 1} \right).\frac{{2x}}{{2.\sqrt {{x^2} + 1} }}}\\{ = 3.\sqrt {{x^2} + 1} + \left( {3x - 1} \right).\frac{x}{{\sqrt {{x^2} + 1} }}}\end{array}\]

Bước 3:

\[\begin{array}{*{20}{l}}{ = \frac{{3.\left( {{x^2} + 1} \right) + 3{x^2} - x}}{{\sqrt {{x^2} + 1} }}}\\{ = \frac{{6{x^2} - x + 3}}{{\sqrt {{x^2} + 1} }}}\end{array}\]

Đáp án cần chọn là: D

\[y\prime = \frac{{(2{x^2} + 3x - 1)\prime ({x^2} - 5x + 2) - (2{x^2} + 3x - 1)({x^2} - 5x + 2)\prime }}{{{{({x^2} - 5x + 2)}^2}}}\]

\[y\prime = \frac{{(4x + 3)({x^2} - 5x + 2) - (2{x^2} + 3x - 1)(2x - 5)}}{{{{({x^2} - 5x + 2)}^2}}}\]

\( = \frac{{4{x^3} - 20{x^2} + 8x + 3{x^2} - 15x + 6 - 4{x^3} - 6{x^2} + 2x + 10{x^2} + 15x - 5}}{{{{({x^2} - 5x + 2)}^2}}}\)

\[y\prime = \frac{{ - 13{x^2} + 10x + 1}}{{{{({x^2} - 5x + 2)}^2}}}\]