What is (\partial)/(\partial z)(x/(x^2+y^2+z^2)) ?

The answer to (\partial)/(\partial z)(x/(x^2+y^2+z^2)) is -(2xz)/((x^2+y^2+z^2)^2)

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(\partial)/(\partial z)(x/(x^2+y^2+z^2))

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

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

Solution steps

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

Treat

x, y

as constants

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d y}{d x}, x^{8} + y^{7} = 5

(e^{3 t})^{^{'}}

\int - \frac{7}{x ^{2}} - 4 d x

\int \frac{3 t ^{4} - t ^{3} + 6 t ^{2}}{t ^{4}} d t

csc (y) d x + sec^{2} (x) d y = 0

What is (\partial)/(\partial z)(x/(x^2+y^2+z^2)) ?
The answer to (\partial)/(\partial z)(x/(x^2+y^2+z^2)) is -(2xz)/((x^2+y^2+z^2)^2)