derivative of (|x|/(sqrt(2-x^2)))

解答

\frac{d}{d x} (\frac{∣ x ∣}{2 - x ^{2}})

\frac{2 x}{∣ x ∣ ( 2 - x ^{2} ) - x ^{2} + 2}

求解步骤

\frac{d}{d x} (\frac{∣ x ∣}{2 - x ^{2}})

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} ( ∣ x ∣ ) 2 - x ^{2} - \frac{d}{d x} ( 2 - x ^{2} ) ∣ x ∣}{( 2 - x ^{2} ) ^{2}}

\frac{d}{d x} (∣ x ∣) = \frac{x}{∣ x ∣}

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

= \frac{\frac{x}{∣ x ∣} 2 - x ^{2} - ( - \frac{x}{2 - x ^{2}} ) ∣ x ∣}{( 2 - x ^{2} ) ^{2}}

化简

\frac{\frac{x}{∣ x ∣} 2 - x ^{2} - ( - \frac{x}{2 - x ^{2}} ) ∣ x ∣}{( 2 - x ^{2} ) ^{2}} : \frac{2 x}{∣ x ∣ ( 2 - x ^{2} ) - x ^{2} + 2}

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

3 (1 + t^{2}) (\frac{d y}{d t}) = 2 t y (y^{3} - 1)

\int \frac{7 ( 4 - x ^{2} )}{( x - 4 ) ( x - 2 ) ( x + 3 )} d x

\int \frac{1}{5} sin (5 x) d x

\frac{d}{d t} (- 4 t^{3})

\int - \frac{1}{8} sin (2 x) d x