derivative of ln|tan(x/2 |)

解答

\frac{d}{d x} (ln tan (\frac{x}{2}))

csc (x)

求解步骤

\frac{d}{d x} (ln tan (\frac{x}{2}))

使用连锁律:

\frac{1}{tan ( \frac{x}{2} )} \frac{d}{d x} (tan (\frac{x}{2}))

= \frac{1}{tan ( \frac{x}{2} )} \frac{d}{d x} (tan (\frac{x}{2}))

\frac{d}{d x} (tan (\frac{x}{2})) = \frac{sec ^{2} ( \frac{x}{2} ) tan ( \frac{x}{2} )}{2 tan ( \frac{x}{2} )}

= \frac{1}{tan ( \frac{x}{2} )} \cdot \frac{sec ^{2} ( \frac{x}{2} ) tan ( \frac{x}{2} )}{2 tan ( \frac{x}{2} )}

化简

\frac{1}{tan ( \frac{x}{2} )} \cdot \frac{sec ^{2} ( \frac{x}{2} ) tan ( \frac{x}{2} )}{2 tan ( \frac{x}{2} )} : csc (x)

= csc (x)

\frac{d}{d x} (\frac{9 - x ^{2}}{32 x})

d er i v a t i v e s (t) = 14 t^{2} + 840 t

s im pl i f y 0

x \to 0 lim (\frac{0 \cdot x ^{6}}{x ^{9}})

3 y^{^{''}} + 75 y = 0