What is (\partial)/(\partial x)((4x^3)/(x^2+y^2)) ?

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

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(\partial)/(\partial x)((4x^3)/(x^2+y^2))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\frac{d y}{d x} = 9 y e^{x + 9}

\frac{\partial}{\partial x} (\frac{1}{2 x})

x \to 8 lim (- (2 x - 8))

\int (7 x^{2} - 8 x) e^{2 x} d x

y^{^{'}} = \frac{- y}{x}

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