Найти производную функции f(x)=(x-4)ctgx

Question

Найти производную функции f(x)=(x-4)ctgx

Answer 1

(u*v)' = u'*v+u*v'
f '(x) = ( (x-4)*ctg(x) )' = (x-4)'*ctg(x) + (x-4)*ctg'(x) = W

(u/v)' = (u'*v - u*v')/(v^2)
ctg'(x) = (cos(x)/sin(x))' = ( (cos(x))'*sin(x) - cos(x)*(sin(x))' )/(sin^2(x) = 
= ( -sin^2(x) - cos^2(x) )/(sin^2(x)) = - 1/(sin^2(x)).

W = ctg(x) + (x-4)*(-1/sin^2(x)) = ctg(x) - ( (x-4)/sin^2(x) ).