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

The answer to (\partial)/(\partial x)(-y/(sqrt(x^2+y^2))) is (yx)/((x^2+y^2)^{3/2)}

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

y = \frac{1}{1 + x ^{2}}, (1, 0.5)

y = x^{3} + 6 x - 2, (1, 5)

\frac{d}{d x} (\frac{9}{6 - x})

\int 2 x^{3} + 3 d x

What is (\partial)/(\partial x)(-y/(sqrt(x^2+y^2))) ?
The answer to (\partial)/(\partial x)(-y/(sqrt(x^2+y^2))) is (yx)/((x^2+y^2)^{3/2)}