What is the derivative of (y(4x+4y))/((x^2+y^2)) ?

The derivative of (y(4x+4y))/((x^2+y^2)) is (-4xy^2+8x^2y+4x^3)/((x^2+y^2)^2)

What is the first derivative of (y(4x+4y))/((x^2+y^2)) ?

The first derivative of (y(4x+4y))/((x^2+y^2)) is (-4xy^2+8x^2y+4x^3)/((x^2+y^2)^2)

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derivative of

\frac{y ( 4 x + 4 y )}{( x ^{2} + y ^{2} )}

\frac{- 4 x y ^{2} + 8 x ^{2} y + 4 x ^{3}}{( x ^{2} + y ^{2} ) ^{2}}

Solution steps

\frac{d}{d y} (\frac{y ( 4 x + 4 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{d}{d y} ( y ( 4 x + 4 y ) ) ( x ^{2} + y ^{2} ) - \frac{d}{d y} ( x ^{2} + y ^{2} ) y ( 4 x + 4 y )}{( x ^{2} + y ^{2} ) ^{2}}

\frac{d}{d y} (y (4 x + 4 y)) = 4 x + 8 y

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

= \frac{( 4 x + 8 y ) ( x ^{2} + y ^{2} ) - 2 yy ( 4 x + 4 y )}{( x ^{2} + y ^{2} ) ^{2}}

Simplify

\frac{( 4 x + 8 y ) ( x ^{2} + y ^{2} ) - 2 yy ( 4 x + 4 y )}{( x ^{2} + y ^{2} ) ^{2}} : \frac{- 4 x y ^{2} + 8 x ^{2} y + 4 x ^{3}}{( x ^{2} + y ^{2} ) ^{2}}

= \frac{- 4 x y ^{2} + 8 x ^{2} y + 4 x ^{3}}{( x ^{2} + y ^{2} ) ^{2}}

\int_{1}^{4} x^{2} + 2 d x

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

\frac{\partial}{\partial t} (- 3 s t)

(0.1666) (0.998)^{x}

\frac{d}{d x} (\frac{2 x ^{2} - 5 x + 4}{x})

What is the derivative of (y(4x+4y))/((x^2+y^2)) ?
The derivative of (y(4x+4y))/((x^2+y^2)) is (-4xy^2+8x^2y+4x^3)/((x^2+y^2)^2)
What is the first derivative of (y(4x+4y))/((x^2+y^2)) ?
The first derivative of (y(4x+4y))/((x^2+y^2)) is (-4xy^2+8x^2y+4x^3)/((x^2+y^2)^2)