derivative of (1+tan^2(x)/(xsec(x)))

解答

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

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

求解步骤

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

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( 1 + tan ^{2} ( x ) ) x sec ( x ) - \frac{d}{d x} ( x sec ( x ) ) ( 1 + tan ^{2} ( x ) )}{( x sec ( x ) ) ^{2}}

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

\frac{d}{d x} (x sec (x)) = sec (x) + x sec (x) tan (x)

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

化简

\frac{2 sec ^{2} ( x ) tan ( x ) x sec ( x ) - ( sec ( x ) + x sec ( x ) tan ( x ) ) ( 1 + tan ^{2} ( x ) )}{( x sec ( x ) ) ^{2}} : \frac{sec ( x ) ( x tan ( x ) - 1 )}{x ^{2}}

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

in v erse l a pl a ce \frac{1}{s ^{2} ( s ^{2} + 3 s + 2 )}

\int sin^{6} (t) cos^{3} (t) d t

x \to 8 lim (8 x + ∣ x - 8 ∣)

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

t an g e n t 2 x y^{3} + x y = 18, (6, 1)