Производные 1. ((sinx+cosx)²)' 2.(1+tg²x/1+ctg²x)' 3.(lnsinx)'

1) (sin x+cos x)²= 1 + sin2x ⇒ (1 + sin2x)' = 2cos2x ⇒ ((sin x+cos x)²)' = 2cos2x

2) $\dfrac{1+tg^2x}{1+ctg^2x} = \dfrac{\frac{1}{cos^2x}}{\frac{1}{sin^2x}} = \dfrac{sin^2x}{cos^2x} =tg^2x$

⇒ $(tg^2x)'=2tgx*\frac{1}{cos^2x} =\frac{2tgx}{cos^2x}$ ⇒ $\left[\begin{array}{c}\dfrac{1+tg^2x}{1+ctg^2x}\end{array}\right] ' = \dfrac{2tgx}{cos^2x}$

3) $(\ln sin\ x )' = \frac{1}{sin\ x} *cos\ x = ctg\ x$