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

The answer to (\partial)/(\partial x)((x^4+y^4)/(x^2+y^2)) is (2x^5+4x^3y^2-2xy^4)/((x^2+y^2)^2)

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Quotient Rule:

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

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

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

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

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

Expand

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

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

\frac{d}{d x} (5 sin^{3} (x))

f (x) = 2 x e^{x}

y^{^{'}} + x (y^{2} + y) = 0

\frac{d}{d x} (3 cos (\frac{x}{2}))

\frac{d y}{d x}, y = - \frac{x}{y}

What is (\partial)/(\partial x)((x^4+y^4)/(x^2+y^2)) ?
The answer to (\partial)/(\partial x)((x^4+y^4)/(x^2+y^2)) is (2x^5+4x^3y^2-2xy^4)/((x^2+y^2)^2)