What is (\partial)/(\partial x)(-y/(4x^2+9y^2)) ?

The answer to (\partial)/(\partial x)(-y/(4x^2+9y^2)) is (8yx)/((4x^2+9y^2)^2)

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(\partial)/(\partial x)(-y/(4x^2+9y^2))

\frac{\partial}{\partial x} (- \frac{y}{4 x ^{2} + 9 y ^{2}})

\frac{8 y x}{( 4 x ^{2} + 9 y ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (- \frac{y}{4 x ^{2} + 9 y ^{2}})

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= - y \frac{\partial}{\partial x} (\frac{1}{4 x ^{2} + 9 y ^{2}})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= - y \frac{\partial}{\partial x} ((4 x^{2} + 9 y^{2})^{- 1})

Apply the chain rule:

- \frac{1}{( 4 x ^{2} + 9 y ^{2} ) ^{2}} \frac{\partial}{\partial x} (4 x^{2} + 9 y^{2})

= - \frac{1}{( 4 x ^{2} + 9 y ^{2} ) ^{2}} \frac{\partial}{\partial x} (4 x^{2} + 9 y^{2})

\frac{\partial}{\partial x} (4 x^{2} + 9 y^{2}) = 8 x

= - y (- \frac{1}{( 4 x ^{2} + 9 y ^{2} ) ^{2}} \cdot 8 x)

Simplify

- y (- \frac{1}{( 4 x ^{2} + 9 y ^{2} ) ^{2}} \cdot 8 x) : \frac{8 y x}{( 4 x ^{2} + 9 y ^{2} ) ^{2}}

= \frac{8 y x}{( 4 x ^{2} + 9 y ^{2} ) ^{2}}

y^{^{'}} = x^{2} y - x^{2} y^{2}, y (0) = 17

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What is (\partial)/(\partial x)(-y/(4x^2+9y^2)) ?
The answer to (\partial)/(\partial x)(-y/(4x^2+9y^2)) is (8yx)/((4x^2+9y^2)^2)