Question

Решите пожалуйста 3 производных

Answer 1

$1)y`=(sin ^{2} 3t)`=2sin3tcdot (sin3t)`=2sin3tcdot cos3tcdot(3t)`=3sin6t, \ 2)y`=(e ^{ctgx})`=e ^{ctgx}cdot (ctgx)`=e ^{ctgx}cdot (- frac{1}{sin ^{2}x }) =-frac{e ^{ctgx}}{sin ^{2}x },$

$3) y`=3ln ^{2}(x+ sqrt{ x^{2} +1})cdot(ln (x+ sqrt{ x^{2} +1}))`= \ =3ln ^{2}(x+ sqrt{ x^{2} +1})cdot frac{1}{x+ sqrt{ x^{2} +1} } cdot(x+ sqrt{ x^{2} +1})` = \=3ln ^{2}(x+ sqrt{ x^{2} +1})cdot frac{1}{x+ sqrt{ x^{2} +1} } cdot(1+ frac{2x}{2 sqrt{ x^{2} +1} } })=$
$=3ln ^{2}(x+ sqrt{ x^{2} +1})cdot frac{1}{x+ sqrt{ x^{2} +1} } cdot( frac{( sqrt{ x^{2} +1}+ x)}{ sqrt{ x^{2} +1} } })= \ =3ln ^{2}(x+ sqrt{ x^{2} +1})cdot ( frac{1}{ sqrt{ x^{2} +1} } })= \ frac{3}{ sqrt{ x^{2} +1} } }cdot ln ^{2}(x+ sqrt{ x^{2} +1})$