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

解答

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

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

求解步骤

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

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

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

使用指数法则:

\frac{1}{a} = a^{- 1}

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

使用连锁律:

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

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

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

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

化简

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

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

\frac{d y}{d x} = \frac{2 y}{3 x}

t an g e n t f (x) = 5 - 3 x, at x = - 2

x \to - 5 lim (4 x^{3} + x)

\frac{d y}{d x} + 3 y = 2

x \to 3 lim (- 8)