Question

Вычислить производную функции √(2+x^2)/sin (x^2)

Answer 1

$( \frac{ \sqrt{2+ x^{2} } }{sin(x^2)} )'=( \frac{ (2+ x^{2})^{ \frac{1}{2} } }{sin(x^2)} )'= \frac{ \frac{1}{2} (2+x^2)^{- \frac{1}{2} }*2x*sin(x^2)-(2+x^2)^{ \frac{1}{2}}*cos(x^2)*2x }{sin^2(x^2)} = \\ =\frac{ \frac{xsin( x^{2} )}{\sqrt {2+x^{2} } }-2x \sqrt {2+x^2}*cos(x^2)}{sin^2(x^2)} =\frac{ \frac{x}{\sqrt {2+ x^{2} } }-2x \sqrt {2+x^2}*ctg(x^2)}{sin(x^2)} =$
$=\frac{ x-2x (2+x^2)*ctg(x^2)}{sin(x^2) \sqrt{2+x^2} }=\frac{ x-2x (2+x^2)*ctg(x^2)}{sin(x^2) (2+x^2) } \sqrt{2+x^2}$