Question

Помогите с решением производной:

Answer 1

$1)\ \ y=arccosx+arcsinx\\\\y'=-\dfrac{1}{\sqrt{1-x^2}}+\dfrac{1}{\sqrt{1-x^2}}=0\\\\\\2)\ \ y=sin5x\cdot cos^2x\\\\y'=5cos5x\cdot cos^2x+sin5x\cdot 2cosx\cdot (-sinx)=5cos5x\cdot cos^2x-sin5x\cdot sin2x]\\\\\\3)\ \ y=ln\Big(\dfrac{1}{3}\, x^3+1\Big)^3\\\\y'=\dfrac{1}{(\frac{1}{3}\, x^3+1)^3}\cdot 3\Big(\dfrac{1}{3}\, x^3+1\Big)^2\cdot \dfrac{1}{3}\cdot 3x^2=\dfrac{3x^2}{(\frac{1}{3}\, x^3+1)^3}\cdot \Big(\dfrac{1}{3}\, x^3+1\Big)^2=\\\\\\=\dfrac{3x^2}{\frac{1}{3}\, x^3+1}=\dfrac{9x^2}{x^3+3}$

$4)\ \ y=\dfrac{4}{x}+arctg\sqrt{x}\\\\y'=-\dfrac{4}{x^2}+\dfrac{1}{1+x}\cdot \dfrac{1}{2\sqrt{x}}=-\dfrac{4}{x^2}+\dfrac{1}{2\sqrt{x}\, (1+x)}$