Question

Найти производную сложной функции : y=(cos2x)^arctg√x

Answer 1

Найти производную сложной функции : y=(cos2x)^arctg√x
ln(y)= ln((cos2x)^arctg√x)
(1/y)·y⁽¹⁾=[ln((cos2x)^arctg√x)]⁽¹⁾

y⁽¹⁾=y·[(arctg√x)·ln(cos2x)]⁽¹⁾
y⁽¹⁾=[(cos2x)^arctg√x]·[{(arctg√x)}⁽¹⁾·ln(cos2x)+(arctg√x)·{ln(cos2x)}⁽¹⁾]

y⁽¹⁾=[(cos2x)^arctg√x]·
·[{1/(1+x)}·(1/(2√x))·ln(cos2x)+(arctg√x)·{1/(cos2x)}·(-sin2x)·2]=
=[(cos2x)^arctg√x]·{ln(cos2x)/(2(√x)(x+1))-2·(sin2x)·(arctg√x)/cos2x}