What is (\partial)/(\partial y)((3x-y)/(5xy+9)) ?

The answer to (\partial)/(\partial y)((3x-y)/(5xy+9)) is (-15x^2-9)/((5xy+9)^2)

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(\partial)/(\partial y)((3x-y)/(5xy+9))

\frac{\partial}{\partial y} (\frac{3 x - y}{5 x y + 9})

\frac{- 15 x ^{2} - 9}{( 5 x y + 9 ) ^{2}}

Solution steps

\frac{\partial}{\partial y} (\frac{3 x - y}{5 x y + 9})

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} ( 3 x - y ) ( 5 x y + 9 ) - \frac{\partial}{\partial y} ( 5 x y + 9 ) ( 3 x - y )}{( 5 x y + 9 ) ^{2}}

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

\frac{\partial}{\partial y} (5 x y + 9) = 5 x

= \frac{( - 1 ) ( 5 x y + 9 ) - 5 x ( 3 x - y )}{( 5 x y + 9 ) ^{2}}

Simplify

\frac{( - 1 ) ( 5 x y + 9 ) - 5 x ( 3 x - y )}{( 5 x y + 9 ) ^{2}} : \frac{- 15 x ^{2} - 9}{( 5 x y + 9 ) ^{2}}

= \frac{- 15 x ^{2} - 9}{( 5 x y + 9 ) ^{2}}

x y^{^{'}} - y = x^{2} sin (x)

\frac{d x}{d y} = 5 x (6 - x)

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

x \to 0 lim (\frac{( 1 + x ) ^{5} - 1}{x})

y \to - 3 lim (lo g_{2} (3) y^{2})

What is (\partial)/(\partial y)((3x-y)/(5xy+9)) ?
The answer to (\partial)/(\partial y)((3x-y)/(5xy+9)) is (-15x^2-9)/((5xy+9)^2)