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

The answer to (\partial)/(\partial x)(((xv^2))/((x^2+v^2))) is (v^2(-x^2+v^2))/((x^2+v^2)^2)

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(\partial)/(\partial x)(((xv^2))/((x^2+v^2)))

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

\frac{v ^{2} ( - x ^{2} + v ^{2} )}{( x ^{2} + v ^{2} ) ^{2}}

Solution steps

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

Treat

v

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

= v^{2} \frac{1 \cdot ( x ^{2} + v ^{2} ) - 2 xx}{( x ^{2} + v ^{2} ) ^{2}}

Simplify

v^{2} \frac{1 \cdot ( x ^{2} + v ^{2} ) - 2 xx}{( x ^{2} + v ^{2} ) ^{2}} : \frac{v ^{2} ( - x ^{2} + v ^{2} )}{( x ^{2} + v ^{2} ) ^{2}}

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

What is (\partial)/(\partial x)(((xv^2))/((x^2+v^2))) ?
The answer to (\partial)/(\partial x)(((xv^2))/((x^2+v^2))) is (v^2(-x^2+v^2))/((x^2+v^2)^2)