d/(d{z)}(({z})/(({x)^2+{y}^2+{z}^2)^{3/2}})

解答

\frac{d}{d z} (\frac{z}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}})

\frac{- 2 z ^{2} + x ^{2} + y ^{2}}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

求解步骤

\frac{d}{d z} (\frac{z}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{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 z} ( z ) ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}} - \frac{d}{d z} ( ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}} ) z}{( ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}} ) ^{2}}

\frac{d}{d z} (z) = 1

\frac{d}{d z} ((x^{2} + y^{2} + z^{2})^{\frac{3}{2}}) = 3 z (x^{2} + y^{2} + z^{2})^{\frac{1}{2}}

= \frac{1 \cdot ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}} - 3 z ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{1}{2}} z}{( ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}} ) ^{2}}

化简

\frac{1 \cdot ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}} - 3 z ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{1}{2}} z}{( ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}} ) ^{2}} : \frac{- 2 z ^{2} + x ^{2} + y ^{2}}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

= \frac{- 2 z ^{2} + x ^{2} + y ^{2}}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

\frac{d}{d x} ((sin (5 x)) (arccsc (2) x^{3}))

x y^{^{'}} = y + 2 x^{3} sin^{2} (\frac{y}{x})

\frac{d}{d x} (x^{2} + 100)

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

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