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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

= 2 y^{2} \frac{1 \cdot ( 5 + x ^{2} y ^{2} ) - 2 y ^{2} xx}{( 5 + x ^{2} y ^{2} ) ^{2}}

Simplify

2 y^{2} \frac{1 \cdot ( 5 + x ^{2} y ^{2} ) - 2 y ^{2} xx}{( 5 + x ^{2} y ^{2} ) ^{2}} : \frac{2 y ^{2} ( 5 - x ^{2} y ^{2} )}{( 5 + x ^{2} y ^{2} ) ^{2}}

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

\int 5 x (x^{2} + 1)^{3} d x

\int_{- 3}^{2} 6 - x - x^{2} d x

\frac{d}{d x} (2 x^{4} + 7 x + 5)

x \to - \infty lim (\frac{7}{x})

\int x (x + 2) d x

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