Giải câu 1 bài 3: Đạo hàm của hàm số lượng giác.
a) \(y = \frac{x-1}{5x-2}\)
\( y'=\frac{\left ( x-1 \right )'.\left ( 5x-2 \right )-\left ( x-1 \right ).\left ( 5x-2 \right )'}{\left ( 5x-2 \right )^{2}}\)
\( =\frac{(5x-2)-\left ( x-1 \right ).5}{\left ( 5x-2 \right )^{2}}\)
\( =\frac{3}{\left ( 5x-2 \right )^{2}}\).
b) \(y = \frac{2x+3}{7-3x}\)
\( y'=\frac{\left ( 2x+3 \right )'.\left ( 7-3x \right )-\left ( 2x+3 \right ).\left ( 7-3x \right )'}{\left ( 7-3x \right )^{2}}\)
\(= \frac{2\left ( 7-3x \right )-\left ( 2x+3 \right ).\left ( -3 \right )}{\left ( 7-3x \right )^{2}}\)
\(= \frac{23}{\left ( 7-3x \right )^{2}}\).
c) \(y = \frac{x^{2}+2x+3}{3-4x}\)
\( y'=\frac{\left ( x^{2}+2x+3 \right )'.\left ( 3-4x \right )-\left ( x^{2} +2x+3\right ).\left ( 3-4x \right )'}{\left ( 3-4x \right )^{2}}\)
\(= \frac{\left ( 2x+2 \right ).\left ( 3-4x \right )-\left ( x^{2}+2x+3 \right ).(-4)}{(3-4x)^{2}}\)
\( =\frac{-2(2x^{2}-3x-9)}{(3-4x)^{2}}\).
d) \(y = \frac{x^{2}+7x+3}{x^{2}-3x}\)
\( y'=\frac{(x^{2}+7x+3)'.(x^{2}-3x)-(x^{2}+7x+3).(x^{2}-3x)'}{(x^{2}-3x)^{2}}\)
\( =\frac{(2x-7).(x^{2}-3x)-(x^{2}+7x+3).(2x-3)}{(x^{2}-3x)^{2}}\)
\( =\frac{-10x^{2}-6x+9}{(x^{2}-3x)^{2}}\).