derivative of ln(sqrt(1+sin^2(x)))

解答

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

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

求解步骤

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

使用连锁律:

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

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

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

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

化简

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

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

x \to - 7 lim (\frac{x ^{2} - 8}{7 - x})

\frac{d}{d x} (- cos (x) sin (x))

\int e^{2 x} cos (2 x) d x

\frac{d}{d x} (arctanh (sin (x)))

\int \frac{x}{( 3 x ^{2} - 4 ) ^{4}} d x