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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int p^{3} d p

y^{^{''}} + 9 y = 2 sin (2 t)

x^{2}, x, 0, 1

\frac{d y}{d x} = y (550 - y)

\int \frac{1}{sin ^{2} ( a ) x} + c d x

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