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

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

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(\partial)/(\partial x)(x/(1+x^2y^2))

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

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

Solution steps

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

Treat

y

as a constant

Apply the Quotient Rule:

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

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

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

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

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

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

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

\int x (x + 7)^{6} d x

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

\frac{d}{d x} (4 x^{2} + 5 x + 3)

\int \frac{4 y}{x} d x

\frac{\partial}{\partial y} (x y e^{- 6 y})

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