Question

Необходимо найти производные функций: 1. y= ctg^5(3x)*tg(8x^2) 2. y=9^arccos(1/(x^8) 3. y=4^arcsin(1/(x^9)) 4. y=sin^4(7x)*tg(3x^8)

Answer 1

$1. y = ctg^4(3x)\cdot tg(8x^2)\\y'=(-1-ctg(3x))\cdot4ctg^3(3x)\cdot3+ctg^4(3x)\cdot(1+tg(8x^2))\cdot16x=\\=16x\cdot ctg^4(3x)\cdot(1+tg(8x^2))-12(1+ctg(3x))\cdot ctg^3(3x)\\\\2.\;y=9^{\arccos\left(\frac1{x^8}\right)}\\y'=9^{\arccos\left(\frac1{x^8}\right)}\cdot\ln9\cdot\left(-\frac1{\sqrt{1-\left(\frac1{x^8}\right)^2}}\right)\cdot\left(-\frac8{x^9}\right)=\frac{8\cdot9^{\arccos\left(\frac1{x^8}\right)}\cdot\ln9}{x^9\cdot\sqrt{1-\frac1{x^{16}}}}$

$3.\;y=4^{\arcsin{\frac1{x^9}}}\\y'=4^{\arcsin{\frac1{x^9}}}\cdot\ln4\cdot\frac1{\sqrt{1-\left(\frac1{x^9}\right)^2}}\cdot\left(-\frac{9}{x^{10}}\right)=-\frac{9\cdot4^{\arcsin{\frac1{x^9}}}\cdot\ln4}{x^{10}\cdot\sqrt{1-\frac1{x^{18}}}}\\\\4.\;y=\sin^4(7x)\cdot tg(3x^8)\\y'=\cos(7x)\cdot4\sin^3(7x)\cdot7\cdot tg(3x^8)+\sin^4(7x)\cdot(1+tg^2(3x^8))\cdot24x^7=\\=28\cos(7x)\sin^3(7x)tg(3x^8)+24x^7\sin(7x)(1+tg^2(3x^8))$