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

Lösung

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

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

Schritte zur Lösung

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

Behandle

y, z

als Konstanten

Entferne die Konstante:

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

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

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

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

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

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} ((x^{2} + y^{2} + z^{2})^{- \frac{3}{2}}) = - \frac{3 x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

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

Vereinfache

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

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

\int 100 - x d x

\frac{d ^{2} y}{d t ^{2}} + 9 y = cos (2 t)

\int 2 x - 7 d x

\frac{d}{d θ} (- sin (θ))

s l o p e in t erce pt (6, - 6), (8, 8)