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

The answer to (\partial)/(\partial y)(y/(sqrt(x^2+y^2))) is (x^2)/((x^2+y^2)sqrt(x^2+y^2))

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Simplify

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

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

\frac{\partial}{\partial x} (x^{3} z + yz)

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

\frac{d}{d x} (\frac{8}{2 x})

\frac{\partial}{\partial y} (e^{c o s (x - 2 y)})

\frac{d}{d x} (ln (1 - 8 x))

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