Question

найти производную сложной функции y=sin(ln^3(arctg(2x))

Answer 1

$y_1=sin(ln^3(arctg(2x))\\y=sin(x)\\y'=(sin(x))'*(x)'\\y'=cos(x)=\ \textgreater \ y_1=cos(ln^3(arctg(2x))(ln^3(arctg(2x)))'\\y_2=ln^3(arctg(2x))\\y=ln(x)\\y'=(ln(x))'\\y'= \frac{1}{x} =\ \textgreater \ y=ln^3(x)\\y'= \frac{3}{x} ln^2(x)=\ \textgreater \ y_2= \frac{3ln^2(arctg(2x)}{arctg(2x)} *(arctg(2x))'\\y_3=arctg(2x)\\y_3=arctg(x)\\y'= \frac{1}{1+x^2} =\ \textgreater \ \\y_3'=(arctg(2x))'\\y_3'= \frac{2}{4x^2+1}=\ \textgreater \ \\y_1= \frac{6cos(ln^3(arctg(2x)))(ln^2(arctg(2x)))}{(4x^2+1)(arctg(2x))}$