What is (\partial)/(\partial y)((8x)/(4x^2+5y^2)) ?

The answer to (\partial)/(\partial y)((8x)/(4x^2+5y^2)) is -(80xy)/((4x^2+5y^2)^2)

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(\partial)/(\partial y)((8x)/(4x^2+5y^2))

\frac{\partial}{\partial y} (\frac{8 x}{4 x ^{2} + 5 y ^{2}})

- \frac{80 x y}{( 4 x ^{2} + 5 y ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial y} (\frac{8 x}{4 x ^{2} + 5 y ^{2}})

Treat

x

as a constant

Take the constant out:

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

= 8 x \frac{\partial}{\partial y} (\frac{1}{4 x ^{2} + 5 y ^{2}})

Apply exponent rule:

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

= 8 x \frac{\partial}{\partial y} ((4 x^{2} + 5 y^{2})^{- 1})

Apply the chain rule:

- \frac{1}{( 4 x ^{2} + 5 y ^{2} ) ^{2}} \frac{\partial}{\partial y} (4 x^{2} + 5 y^{2})

= - \frac{1}{( 4 x ^{2} + 5 y ^{2} ) ^{2}} \frac{\partial}{\partial y} (4 x^{2} + 5 y^{2})

\frac{\partial}{\partial y} (4 x^{2} + 5 y^{2}) = 10 y

= 8 x (- \frac{1}{( 4 x ^{2} + 5 y ^{2} ) ^{2}} \cdot 10 y)

Simplify

8 x (- \frac{1}{( 4 x ^{2} + 5 y ^{2} ) ^{2}} \cdot 10 y) : - \frac{80 x y}{( 4 x ^{2} + 5 y ^{2} ) ^{2}}

= - \frac{80 x y}{( 4 x ^{2} + 5 y ^{2} ) ^{2}}

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What is (\partial)/(\partial y)((8x)/(4x^2+5y^2)) ?
The answer to (\partial)/(\partial y)((8x)/(4x^2+5y^2)) is -(80xy)/((4x^2+5y^2)^2)