Ответ:

[tex]y = x {}^{3} \sin(2x) [/tex]

[tex]y = \frac{d}{dx} (x {}^{3} \times \sin(2x) )[/tex]

[tex] \frac{d}{dx} (f \times g) = \frac{d}{dx} (f) \times g + f \times \frac{d}{dx} (g)[/tex]

[tex]y = \frac{d}{dx} (x {}^{3} ) \times \sin(2x) + x {}^{3} \times \frac{d}{dx} ( \sin(2x) )[/tex]

[tex]y = 3x {}^{2} \times \sin(2x) + x {}^{3} \times \frac{d}{dx} ( \sin(2x) )[/tex]

[tex]y = 3x {}^{2} \times \sin(2x) + x {}^{3} \times \cos(2x) \times 2[/tex]

[tex]y = 3x {}^{2} \times \sin(2x) + 2x {}^{3} \times \cos(2x) [/tex]

Verified answer

[tex]\displaystyle\bf\\y=x^{3} Sin2x[/tex]

Формулы производных , которые будут применены в данном задании:

[tex]\displaystyle\bf\\1)\\\\(x^{n} )'=n\cdot x^{n-1} \\\\2)\\\\(Sinx)'=Cosx\\\\3)\\\\(u\cdot v)'=u'\cdot v+u\cdot v'\\\\4)\\\\(Sinu)'=Cosu\cdot u'\\\\\\y'=(x^{3} )'\cdot Sin2x+x^{3} \cdot (Sin2x)'=3x^{2} \cdot Sin2x+x^{3} \cdot 2Cos2x\\\\\\Otvet \ : \ y'=3x^{2} Sin2x+2x^{3} Cos2x[/tex]