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Bài 3. Tìm đạo hàm của các hàm số sau:
a) \(y = 5sinx -3cosx\);
b) \( y=\frac{sinx+cosx}{sinx-cosx}\);
c) \(y = x cotx\);
d) \(y = \frac{sinx}{x}\) + \( \frac{x}{sinx}\);
e) \(y = \sqrt{(1 +2tan x)}\);
f) \(y = sin\sqrt{(1 +x^2)}\).
a) \(y’=5cosx-3(-sinx)=5cosx+3sinx\);
b) \( y’={{(sinx+cos x)’.(sin x- cos x)-(sin x+cos x)(sin x-cos x)’}\over{(sin x-cos x)^{2}}}\) = \( {{(cos x-sin x)(sin x -cos x)-(sin x+ cos x)(cosx+sinx)}\over{(sin x-cosx )^{2}}}\) = \( {{-2}\over{(sin x-cos x)^{2}}}\).
c) \(y’ = cotx +x. \left ( -\frac{1}{sin^{2}x} \right )= cotx – \frac{x}{sin^{2}x}\).
d) \( y’=\frac{(sin x)’.x-sin x.(x)’}{x^{2}}\) +\( \frac{(x)’.sin x-x(sin x)’}{sin^{2}x}\) = \( \frac{x.cosx-sinx}{x^{2}}+\frac{sin x-x.cosx}{sin^{2}x}\)\( = (x. cosx -sinx) \left ( \frac{1}{x^{2}}-\frac{1}{sin^{2}x} \right )\).
e) \( y’=\frac{(1+2tanx)’}{2\sqrt{1+2tanx}}\) = \( \frac{\frac{2}{cos^{2}x}}{2\sqrt{1+2tanx}}\) = \( \frac{1}{cos^{2}x\sqrt{1+2tanx}}\).
f) \(y’ = (\sqrt{(1+x^2)})’ cos\sqrt{(1+x^2)} \)\(= \frac{(1+x^{2})’}{2\sqrt{1+x^{2}}}cos\sqrt{(1+x^2)} = \frac{x}{\sqrt{1+x^{2}}}cos\sqrt{(1+x^2)}\).