その他の三角函数の微分

《review》

(sin x)' = cos x, (cos x)' = -sin x.
tan x = (sin x)/(cos x),
[商の微分公式] (f(x)/g(x))' = (f'(x)g(x) - f(x)g'(x))/g(x)².

上記の公式を用いると

(tan x)' = ((sin x)/(cos x))' = ((sin x)'cos x - sin x (cos x)')/cos² x
(cos² x - sin x (-sin x))/cos² x = (cos² x + sin² x)/cos² x
= 1/cos² x = sec² x.

(sec x)' = (1/cos x)' = -(cos x)'/cos² x = sin x/cos² x = tan x sec x.

(csc x)' = (1/sin x)' = -(sin x)'/sin² x = -cos x/sin² x = -cot x csc x.

(cot x)' = (cos x/sin x)' = ((cos x)'sin x - cos x(sin x)')/sin² x
=(-sin²x - cos²x)/sin² x = -(sin²x + cos²x)/sin² x
= -1/sin² x = -csc²x.

次へ
微分 2 の目次へ