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

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

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

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

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

Solution steps

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

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

Simplify

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

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

\int 2 x 1 - x^{2} d x

\int \frac{1}{- x ^{2} - 8 x - 7} d x

\frac{d}{d x} ((sec (x))^{2})

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

\frac{d}{d x} (c \cdot ln (x))

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