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Bài 4. Tìm đạo hàm của các hàm số sau:
a) \(y = x^2 – x\sqrt x + 1\);
b) \(y = \sqrt {(2 – 5x – x^2)}\);
c) \(y = \frac{x^{3}}{\sqrt{a^{2}-x^{2}}}\) ( \(a\) là hằng số);
d) \(y = \frac{1+x}{\sqrt{1-x}}\).
a) \(y’ = 2x – \left ( \sqrt{x}+x.\frac{1}{2\sqrt{x}} \right )\) \(= 2x – \frac{3}{2}\sqrt{x}\).
b) \(y’ =\frac{\left ( 2-5x-x^{2} \right )’}{2.\sqrt{2-5x-x^{2}}}\) = \( \frac{-5-2x}{2\sqrt{2-5x-x^{2}}}\).
c) \(y’ = \frac{\left ( x^{3} \right )’.\sqrt{a^{2}-x^{2}}-x^{3}.\left ( \sqrt{a^{2}-x^{2}} \right )}{a^{2}-x^{2}}\) = \( \frac{3x^{2}.\sqrt{a^{2}-x^{2}}-x^{3}.\frac{-2x}{2\sqrt{a^{2}-x^{2}}}}{a^{2}-x^{2}}\) = \( \frac{3x^{2}.\sqrt{a^{2}-x^{2}}+\frac{x^{4}}{\sqrt{a^{2}-x^{2}}}}{a^{2}-x^{2}}\) = \( \frac{x^{2}\left ( 3a^{2}-2x^{2} \right )}{\left ( a^{2} -x^{2}\right )\sqrt{a^{2}-x^{2}}}\).
d) \(y’ = \frac{\left ( 1+x \right )’.\sqrt{1-x}-\left ( 1+x \right ).\left ( \sqrt{1-x} \right )’}{1-x}\) = \( \frac{\sqrt{1-x}-\left ( 1+x \right )\frac{-1}{2\sqrt{1-x}}}{1-x}\) = \( \frac{2\left ( 1-x \right )+1+x}{2\left ( 1-x \right )\sqrt{1-x}}\) = \( \frac{3-x}{2\left ( 1-x \right )\sqrt{1-x}}\).