derivative f(x)=arctan(sin(4x))

解

導関数

f (x) = arctan (sin (4 x))

\frac{4 cos ( 4 x )}{sin ^{2} ( 4 x ) + 1}

解答ステップ

\frac{d}{d x} (arctan (sin (4 x)))

連鎖法則を適用:

\frac{1}{( sin ( 4 x ) ) ^{2} + 1} \frac{d}{d x} (sin (4 x))

= \frac{1}{( sin ( 4 x ) ) ^{2} + 1} \frac{d}{d x} (sin (4 x))

\frac{d}{d x} (sin (4 x)) = cos (4 x) \cdot 4

= \frac{1}{( sin ( 4 x ) ) ^{2} + 1} cos (4 x) \cdot 4

簡素化

\frac{1}{sin ^{2} ( 4 x ) + 1} cos (4 x) \cdot 4 : \frac{4 cos ( 4 x )}{sin ^{2} ( 4 x ) + 1}

= \frac{4 cos ( 4 x )}{sin ^{2} ( 4 x ) + 1}