Ответ:
Производная дроби равна [tex]\bf \Big(\dfrac{u}{v}\Big)'=\dfrac{u'v-uv'}{v^2}[/tex] .
[tex]\bf (u^3)'=3u^2\cdot u'\ \ ,\ \ \ (arcsinu)'=\dfrac{1}{\sqrt{1-u^2}}\cdot u'\ \,\ \ \ (sinu)'=cosu\cdot u'[/tex] .
[tex]y=\dfrac{arcsin^34x}{sin(3x+1)}\\\\\\y'=\dfrac{3\, arcsin^24x\cdot (arcsin4x)'\cdot sin(3x+1)-arcsin^34x\cdot cos(3x+1)\cdot (3x+1)'}{sin^2(3x+1)}=\\\\\\=\dfrac{3\, arcsin^24x\cdot \dfrac{1}{\sqrt{1-16x^2}}\cdot 4\cdot sin(3x+1)-arcsin^34x\cdot cos(3x+1)\cdot 3}{sin^2(3x+1)}=\\\\\\=\dfrac{12\, arcsin^24x\cdot sin(3x+1)-3\, \sqrt{1-16x^2}\cdot arcsin^34x\cdot cos(3x+1)}{\sqrt{1-16x^2}\cdot sin^2(3x+1)}[/tex]
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Answers & Comments
Ответ:
Производная дроби равна [tex]\bf \Big(\dfrac{u}{v}\Big)'=\dfrac{u'v-uv'}{v^2}[/tex] .
[tex]\bf (u^3)'=3u^2\cdot u'\ \ ,\ \ \ (arcsinu)'=\dfrac{1}{\sqrt{1-u^2}}\cdot u'\ \,\ \ \ (sinu)'=cosu\cdot u'[/tex] .
[tex]y=\dfrac{arcsin^34x}{sin(3x+1)}\\\\\\y'=\dfrac{3\, arcsin^24x\cdot (arcsin4x)'\cdot sin(3x+1)-arcsin^34x\cdot cos(3x+1)\cdot (3x+1)'}{sin^2(3x+1)}=\\\\\\=\dfrac{3\, arcsin^24x\cdot \dfrac{1}{\sqrt{1-16x^2}}\cdot 4\cdot sin(3x+1)-arcsin^34x\cdot cos(3x+1)\cdot 3}{sin^2(3x+1)}=\\\\\\=\dfrac{12\, arcsin^24x\cdot sin(3x+1)-3\, \sqrt{1-16x^2}\cdot arcsin^34x\cdot cos(3x+1)}{\sqrt{1-16x^2}\cdot sin^2(3x+1)}[/tex]