Giải Bài tập 29 trang 108 sgk Toán 11 tập 2 Kết nối

a) $y=3x^{2}-2 \sqrt{x}$

$y' = (3x^2)' - (2\sqrt{x})'$

$y' = 6x - \frac{1}{\sqrt{x}}$

b) $y=\sqrt{1+2x-x^{2}}$

$y' = \left(\sqrt{1+2x-x^2}\right)'$

$y' = \left(\sqrt{u}\right)' = \frac{1}{2\sqrt{u}}(u)'$

$y' = \frac{1}{2\sqrt{1+2x-x^2}}(1+2x-x^2)'$

$(1+2x-x^2)' = 2 - 2x$

$y' = \frac{1}{2\sqrt{1+2x-x^2}}(2 - 2x)$

$y' = \frac{2 - 2x}{2\sqrt{1+2x-x^2}}$

c) $y=tan \frac{x}{2}- cot \frac{x}{2}$

$y' = \left(\tan\frac{x}{2} - \cot\frac{x}{2}\right)'$$y' = \left(\tan\frac{x}{2} - \cot\frac{x}{2}\right)'$

$y' = \left(\tan\frac{x}{2}\right)' - \left(\cot\frac{x}{2}\right)'$

$y' = \frac{1}{\cos^2\frac{x}{2}}\cdot\frac{1}{2} - \frac{-1}{\sin^2\frac{x}{2}}\cdot\frac{1}{2}$

$y' = \frac{1}{2\cos^2\frac{x}{2}} + \frac{1}{2\sin^2\frac{x}{2}}$

d) $y = e^{ex} + \ln(x^2)$

$y' = (e^{ex} + \ln(x^2))'$

$y' = (e^{ex})' + (\ln(x^2))'$

$(e^{ex})' = e^{ex}\cdot(ex)'$

$(e^{ex})' = e^{ex}\cdot(e)'x + e^{ex}\cdot x'$

$(e^{ex})' = e^{ex}\cdot(e)\cdot x + e^{ex}\cdot 1$

$(e^{ex})' = e^{ex}\cdot(ex + 1)$

$(\ln(x^2))' = \frac{1}{x^2}\cdot(x^2)'$

$(\ln(x^2))' = \frac{1}{x^2}\cdot(2x)$

$(\ln(x^2))' = \frac{2x}{x^2}$

$y' = e^{ex}\cdot(ex + 1) + \frac{2x}{x^2}$

$y' = e^{ex}\cdot(ex + 1) + \frac{2}{x}$