What is (\partial)/(\partial x)((-x-6y)/(y^2+5x^2)) ?

The answer to (\partial)/(\partial x)((-x-6y)/(y^2+5x^2)) is (-y^2+5x^2+60xy)/((y^2+5x^2)^2)

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(\partial)/(\partial x)((-x-6y)/(y^2+5x^2))

\frac{\partial}{\partial x} (\frac{- x - 6 y}{y ^{2} + 5 x ^{2}})

\frac{- y ^{2} + 5 x ^{2} + 60 x y}{( y ^{2} + 5 x ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{- x - 6 y}{y ^{2} + 5 x ^{2}})

Treat

y

as a constant

Apply the Quotient Rule:

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

= \frac{\frac{\partial}{\partial x} ( - x - 6 y ) ( y ^{2} + 5 x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + 5 x ^{2} ) ( - x - 6 y )}{( y ^{2} + 5 x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (- x - 6 y) = - 1

\frac{\partial}{\partial x} (y^{2} + 5 x^{2}) = 10 x

= \frac{( - 1 ) ( y ^{2} + 5 x ^{2} ) - 10 x ( - x - 6 y )}{( y ^{2} + 5 x ^{2} ) ^{2}}

Simplify

\frac{( - 1 ) ( y ^{2} + 5 x ^{2} ) - 10 x ( - x - 6 y )}{( y ^{2} + 5 x ^{2} ) ^{2}} : \frac{- y ^{2} + 5 x ^{2} + 60 x y}{( y ^{2} + 5 x ^{2} ) ^{2}}

= \frac{- y ^{2} + 5 x ^{2} + 60 x y}{( y ^{2} + 5 x ^{2} ) ^{2}}

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

\frac{1}{s ^{2} + 4}

7 t \cdot \frac{d y}{d t} - 4 y = t

\frac{d}{d x} (1 - sin (3 x))

What is (\partial)/(\partial x)((-x-6y)/(y^2+5x^2)) ?
The answer to (\partial)/(\partial x)((-x-6y)/(y^2+5x^2)) is (-y^2+5x^2+60xy)/((y^2+5x^2)^2)