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

The answer to (\partial)/(\partial x)((-ky)/(x^2+y^2)) is (2kyx)/((x^2+y^2)^2)

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(\partial)/(\partial x)((-ky)/(x^2+y^2))

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

\frac{2 k y x}{( x ^{2} + y ^{2} ) ^{2}}

Solution steps

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

Simplify

\frac{- k y}{x ^{2} + y ^{2}} : - \frac{k y}{x ^{2} + y ^{2}}

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

Treat

k, y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

5 - 3 t

x^{^{'}} + 2 x = e^{- 2 t} cos (t)

\int y ln (y) d y

f (x) = x + 1

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

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