What is (\partial)/(\partial y)((100)/(5x^2+y^2)) ?

The answer to (\partial)/(\partial y)((100)/(5x^2+y^2)) is -(200y)/((5x^2+y^2)^2)

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(\partial)/(\partial y)((100)/(5x^2+y^2))

\frac{\partial}{\partial y} (\frac{100}{5 x ^{2} + y ^{2}})

- \frac{200 y}{( 5 x ^{2} + y ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial y} (\frac{100}{5 x ^{2} + y ^{2}})

Treat

x

as a constant

Take the constant out:

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

= 100 \frac{\partial}{\partial y} (\frac{1}{5 x ^{2} + y ^{2}})

Apply exponent rule:

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

= 100 \frac{\partial}{\partial y} ((5 x^{2} + y^{2})^{- 1})

Apply the chain rule:

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

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

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

= 100 (- \frac{1}{( 5 x ^{2} + y ^{2} ) ^{2}} \cdot 2 y)

Simplify

100 (- \frac{1}{( 5 x ^{2} + y ^{2} ) ^{2}} \cdot 2 y) : - \frac{200 y}{( 5 x ^{2} + y ^{2} ) ^{2}}

= - \frac{200 y}{( 5 x ^{2} + y ^{2} ) ^{2}}

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

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What is (\partial)/(\partial y)((100)/(5x^2+y^2)) ?
The answer to (\partial)/(\partial y)((100)/(5x^2+y^2)) is -(200y)/((5x^2+y^2)^2)