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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Simplify

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

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

y = - 2 x^{3}, (2, - 16)

x \to 3 lim (\frac{∣ x + 3 ∣}{x - 3})

\int y + 1 d y

\int_{- 5}^{0} 4 + 25 - x^{2} d x

x y^{^{'}} + y + 8 = 0

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