What is (\partial)/(\partial x)((2x^2y)/(5+x^2y^2)) ?

The answer to (\partial)/(\partial x)((2x^2y)/(5+x^2y^2)) is (20xy)/((5+x^2y^2)^2)

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(\partial)/(\partial x)((2x^2y)/(5+x^2y^2))

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

\frac{20 x y}{( 5 + x ^{2} y ^{2} ) ^{2}}

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

2 y \frac{2 x ( 5 + x ^{2} y ^{2} ) - 2 y ^{2} x x ^{2}}{( 5 + x ^{2} y ^{2} ) ^{2}} : \frac{20 x y}{( 5 + x ^{2} y ^{2} ) ^{2}}

= \frac{20 x y}{( 5 + x ^{2} y ^{2} ) ^{2}}

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

ln (5 + x)

\frac{\partial}{\partial x} (8 y^{5} + 8 y^{2} x^{7})

\frac{d}{d x} (\frac{20}{x + 1})

x \to + 0 lim (\frac{7}{x} - \frac{7}{x ^{2} - x})

What is (\partial)/(\partial x)((2x^2y)/(5+x^2y^2)) ?
The answer to (\partial)/(\partial x)((2x^2y)/(5+x^2y^2)) is (20xy)/((5+x^2y^2)^2)