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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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