What is the d/(dy)(y/((x^2+y^2)^{3/2)}) ?

The d/(dy)(y/((x^2+y^2)^{3/2)}) is (-2y^2+x^2)/((x^2+y^2)^{5/2)}

What is the first d/(dy)(y/((x^2+y^2)^{3/2)}) ?

The first d/(dy)(y/((x^2+y^2)^{3/2)}) is (-2y^2+x^2)/((x^2+y^2)^{5/2)}

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

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

Solution steps

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

Treat

x

as a constant

Apply the Quotient Rule:

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

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

\frac{d y}{d y} = 1

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

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

Simplify

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

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

What is the d/(dy)(y/((x^2+y^2)^{3/2)}) ?
The d/(dy)(y/((x^2+y^2)^{3/2)}) is (-2y^2+x^2)/((x^2+y^2)^{5/2)}
What is the first d/(dy)(y/((x^2+y^2)^{3/2)}) ?
The first d/(dy)(y/((x^2+y^2)^{3/2)}) is (-2y^2+x^2)/((x^2+y^2)^{5/2)}