Question

Помогите 100б........

Answer 1

$1. ~ ~y = 3\sin ^2 (3-x^2)$

$\bullet ~ \Big(f(g(x)) \Big) ' = f'(g(x)) \cdot g'(x) \\\\ \bullet ~ (\sin x) ' = \cos x$

$y ' = (3\sin ^2 (3-x^2))' = 3 \cdot \Big ( \big(\sin(3-x^2)\big)^2\Big) '= \\\\ =3\cdot 2 \sin (3-x^2) \cdot (\sin (3-x^2) ) ' = \\\\ = 3 \cdot \underbrace{2 \sin (3-x^2)\cos(3-x^2)}_{\sin (6 -2x^2)} \cdot (-2x) = -6x\cdot \sin (6-2x^2)$

                                                   

$2. ~ y = \log _2(x^2 -2x)$

$\bullet ~ (f(g(x)))' = f'(g(x))g'(x)\\\\\\ \bullet ~~( \log _a u ) ' = \dfrac{u'}{u \ln a }$

$\displaystyle y' = (\log _2(x^2 -2x)) = \frac{1}{(x^2 -2x)\ln 2 } \cdot (x^2 -2x)' = \frac{2x -2}{(x^2 -2x)\ln 2}$

                                                   

$3. ~ y = 2\cos ^3 5x$

$y '=( 2\cos ^3 5x) ' = (2 \cdot (\cos 5x)^3 )' = 2 \cdot 3 \cos ^25x \cdot (\cos 5x)'\cdot (5x)' = \\\\ = 30 \cos^ 5x \cdot (-\sin5x) = - 30 \cos ^2 5x\cdot \sin 5x$

                                           

$4. ~ y = e^{\sqrt{x} }$

$\bullet ~~ (e^u) ' = u'\cdot e^u$

$y ' = (e^{\sqrt{x} }) ' = e^{\sqrt{x} } \cdot (\sqrt{x}) ' = \dfrac{e^{\sqrt{x} }}{2\sqrt{x} }$

                                             

$5. ~ y = \ln ^3 \sin 2x$

$\bullet ~ (f(g(x)))' = f'(g(x))g'(x)\\\\\\ \bullet ~ (\ln u ) '= \dfrac{u'}{u}$

$\displaystyle y' =( \ln ^3 \sin 2x) ' = \Big (\big(\ln ( \sin 2x)\big)^3 \Big )' = 3 \ln^2 (\sin 2x)\cdot (\ln ( \sin 2x)\big)' = \\\\\\ =3 \ln ^2 (\sin 2x)\cdot \frac{(\sin 2x)'}{\sin 2x} = \frac{3\ln ^2 (\sin 2x) \cdot 2 \cos 2x}{\sin2 x} = \frac{6 \ln ^2 (\sin 2x)\cdot \cos 2x}{\sin 2x} = \\\\\\ = 6 \ln ^2 (\sin 2x)\cdot \text{ctg}~2x$