Вычислите производные функции 1) f(x)=cosx*(cosx-1) 2) f(x)=sinx*(ctgx-1)

Вычислим производные функций: 

1)  f (x) = cos x * (cos x - 1);  

f \' (x) = (cos x * (cos x - 1)) \' = cos \' x * (cos x - 1) + (cos x - 1) \' * cos x = -sin x * (cos x - 1) + (-sinx - 0) * cos x = -sin x * (cos x - 1) - sin x * cos x = - sin x * cos x + sin x - sin x * cos x = - 2 * sin x * cos x + sin x = sin x * (1 - 2 * cos x); 

2) f (x) = sin x * (ctg x - 1);  

f \' (x) = (sin x * (ctg x - 1)) \' = cos x * (ctg x - 1) + (ctg x - 1) \' * sin x = cos x * cos x/sin x - cos x + sin x * (-1/sin^2 x) = cos^2x/sin x - cos x - 1/sin x = (cos^2 x - cos x * sin x - 1)/sin x = (cos^2 x - cos x * sin x - sin^2 x - cos^2 x)/sin x = (-cos x * sin x - sin^2 x)/sin x = -sin x * (cos x + sin x)/sin x = - cos x - sin x.