Tính đạo hàm của mỗi hàm số sau
a. \(y = {{2x + 3} \over {{x^2} – 5x + 5}}\)
b. \(y = {1 \over {{{\left( {{x^2} – x + 1} \right)}^5}}}\)
c. \(y = {x^2} + x\sqrt x + 1\)
d. \(y = \left( {x + 1} \right){\left( {x + 2} \right)^2}{\left( {x + 3} \right)^3}\)
e. \(y = \sqrt {{{{x^2} + 1} \over x}} \)
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a. \(y’ = {{2\left( {{x^2} – 5x + 5} \right) – \left( {2x + 3} \right)\left( {2x – 5} \right)} \over {{{\left( {{x^2} – 5x + 5} \right)}^2}}} = {{ – 2{x^2} – 6x + 25} \over {{{\left( {{x^2} – 5x + 5} \right)}^2}}}\)
b. \(y’ = {{ – 5{{\left( {{x^2} – x + 1} \right)}^4}\left( {2x – 1} \right)} \over {{{\left( {{x^2} – x + 1} \right)}^{10}}}} = {{ – 5\left( {2x – 1} \right)} \over {{{\left( {{x^2} – x + 1} \right)}^6}}}\)
c. \(y’ = 2x + \sqrt x + x.{1 \over {2\sqrt x }} = 2x + {3 \over 2}\sqrt x \)
d.
\(\eqalign{ & y’ = {\left( {x + 2} \right)^2}{\left( {x + 3} \right)^2} + \left( {x + 1} \right).2\left( {x + 2} \right){\left( {x + 3} \right)^3} + \left( {x + 1} \right){\left( {x + 2} \right)^2}3{\left( {x + 3} \right)^2} \cr & = 2\left( {x + 2} \right){\left( {x + 3} \right)^2}\left( {3{x^2} + 11x + 9} \right) \cr} \)
e. \(y = \sqrt {x + {1 \over x}} \Rightarrow y’ = {{1 – {1 \over {{x^2}}}} \over {2\sqrt {x + {1 \over x}} }} = {{{x^2} – 1} \over {2{x^2}\sqrt {{{{x^2} + 1} \over x}} }}\)