Question

найти производные заданных функций
1)$y=arctg frac{ sqrt{1+ x^{2} }-1 }{x}$
2)$y=log _{4} (1+sin ^{2} x)$

Answer 1

1) $y'= frac{1}{1+ (frac{ sqrt{1+x^2}-1 }{x} )^2} * frac{ frac{x}{ sqrt{1+x^2} }- sqrt{1+x^2}+1}{x^2} =$
$frac{1}{1+ (frac{ sqrt{1+x^2}-1 }{x} )^2} * frac{{x-x^2}}{sqrt{1+x^2}*x^2}=frac{1}{1+ (frac{ sqrt{1+x^2}-1 }{x} )^2} * frac{{1-x}}{sqrt{1+x^2}*x}$
2) $y'= frac{2sinxcosx}{(1+sin^2x)*ln4} =frac{sin2x}{(1+sin^2x)*ln4}$