What is (\partial)/(\partial x)((xy)/(sqrt(1+xy^2))) ?

The answer to (\partial)/(\partial x)((xy)/(sqrt(1+xy^2))) is (y(2+xy^2))/(2(1+xy^2)sqrt(1+xy^2))

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(\partial)/(\partial x)((xy)/(sqrt(1+xy^2)))

\frac{\partial}{\partial x} (\frac{x y}{1 + x y ^{2}})

\frac{y ( 2 + x y ^{2} )}{2 ( 1 + x y ^{2} ) 1 + x y ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{x y}{1 + x y ^{2}})

Treat

y

as a constant

Take the constant out:

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

= y \frac{\partial}{\partial x} (\frac{x}{1 + x 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 ) 1 + x y ^{2} - \frac{\partial}{\partial x} ( 1 + x y ^{2} ) x}{( 1 + x y ^{2} ) ^{2}}

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

\frac{\partial}{\partial x} (1 + x y^{2}) = \frac{y ^{2}}{2 1 + y ^{2} x}

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

Simplify

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

= \frac{y ( 2 + x y ^{2} )}{2 ( 1 + x y ^{2} ) 1 + x y ^{2}}

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What is (\partial)/(\partial x)((xy)/(sqrt(1+xy^2))) ?
The answer to (\partial)/(\partial x)((xy)/(sqrt(1+xy^2))) is (y(2+xy^2))/(2(1+xy^2)sqrt(1+xy^2))