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Bài 3. Tìm đạo hàm của các hàm số sau:
a) \(y = {({x^{7}} – 5{x^2})^3}\);
b)\(y = ({x^2} + 1)(5 – 3{x^2})\);
c) \(y = \frac{2x}{x^{2}-1}\);
d) \(y = \frac{3-5x}{x^{2}-x+1}\);
e) \(y = \left ( m+\frac{n}{x^{2}} \right )^{3}\) (\(m, n\) là các hằng số).
a) \(y’ = 3.{({x^7} – 5{x^2})^2}.({x^7} – 5{x^2})’ = 3.{({x^{7}} – 5{x^2})^2}.(7{x^6} – 10x)\)
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\(= 3x.{({x^{7}} – 5{x^2})^2}(7{x^5} – 10).\)
b) \(y = 5{x^2} – 3{x^4} + 5 – 3{x^2} = – 3{x^4} + 2{x^2} + 5\), do đó \(y’ = – 12{x^3} + 4x = – 4x.(3{x^2} – 1)\).
c) \(y’ = \frac{\left ( 2x \right )’.\left ( x^{2}-1 \right )-2x\left ( x^{2}-1 \right )’}{\left ( x^{2}-1 \right )^{2}}\) = \( \frac{2.\left ( x^{2}-1 \right )-2x.2x}{\left ( x^{2}-1 \right )^{2}}\) = \( \frac{-2\left ( x^{2}+1 \right )}{\left ( x^{2}-1 \right )^{2}}\).
d) \(y’ = \frac{\left ( 3-5x \right )’\left ( x^{2}-x+1 \right )-\left ( 3-5x \right ).\left ( x^{2}-x+1 \right )’}{\left ( x^{2}-x+1 \right )^{2}}\) = \( \frac{-5\left ( x^{2}-x+1 \right )-\left ( 3-5x \right ).\left ( 2x-1 \right )}{\left ( x^{2}-x+1 \right )^{2}}\) = \( \frac{5x^{2}-6x-2}{\left ( x^{2}-x+1 \right )^{2}}\).
e) \(y’ = 3. \left ( m+\frac{n}{x^{2}} \right )^{2}\) .\( \left ( m+\frac{n}{x^{2}} \right )’\) = 3.\( \left ( m+\frac{n}{x^{2}} \right )^{2}\) \( \left ( -\frac{2n}{x^{3}} \right )\) = -\( \frac{6n}{x^{3}}\) .\( \left ( m+\frac{n}{x^{2}} \right )^{2}\).