derivative of x/(arctan(x))

解答

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

\frac{arctan ( x ) ( x ^{2} + 1 ) - x}{arctan ^{2} ( x ) ( x ^{2} + 1 )}

求解步骤

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

使用微分除法定则:

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

= \frac{\frac{d x}{d x} arctan ( x ) - \frac{d}{d x} ( arctan ( x ) ) x}{( arctan ( x ) ) ^{2}}

\frac{d x}{d x} = 1

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

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

化简

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

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

im pl i c i t \frac{d y}{d x}, y sin (\frac{1}{y}) = 1 - x y

\frac{d}{d x} (lo g_{8} (x))

\int 7 e^{- 0.5 x} d x

7 y^{^{''}} + 13 y^{^{'}} - 2 y = 0

s im pl i f y 2 t^{- \frac{1}{4}}