Question

Найти производную
(tg(x))^4e^x

Answer 1

По формуле:

$y' = ( ln(y))' \times y$

$( ln(y))' = ( ln( {tg(x)}^{4 {e}^{x} } )' = (4 {e}^{x} \times ln(tg(x)) )' = 4 {e}^{x} ln(tg(x)) + \frac{1}{tg(x)} \times \frac{1}{ { \cos(x) }^{2} } \times 4 {e}^{x} = 4 {e}^{x} ( ln(tg(x)) + \frac{ \cos(x) }{ \sin(x) } \times \frac{1}{ { \cos(x) }^{2} } ) = 4 {e}^{x} ( ln(tg(x)) + \frac{1}{ \sin(x) \cos(x) } )$

$y' = {tg(x)}^{4 {e}^{x} } \times4 {e}^{x} ( ln(tg(x)) + \frac{1}{ \sin(x) \cos(x) } )$