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