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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

\frac{\partial}{\partial x} (x^{4} + y^{2}) = 4 x^{3}

= y \frac{2 x ( x ^{4} + y ^{2} ) - 4 x ^{3} x ^{2}}{( x ^{4} + y ^{2} ) ^{2}}

Simplify

y \frac{2 x ( x ^{4} + y ^{2} ) - 4 x ^{3} x ^{2}}{( x ^{4} + y ^{2} ) ^{2}} : \frac{y ( - 2 x ^{5} + 2 x y ^{2} )}{( x ^{4} + y ^{2} ) ^{2}}

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

∣ x - 2 ∣, 0 = x - 3 y + 6

\int_{2}^{3} \frac{40}{3 - x} d x

\frac{d}{d x} (\frac{cos ( x )}{1 + a sin ( x )})

\int \frac{6}{1 + x ^{2}} d x

x \to 1 lim (f (x) + g (x))

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