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

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

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

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

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

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

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

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

x \to 1 lim (- 2 x^{2} + 2 k)

\int 2^{c o s (x)} \cdot sin (x) d x

\int ln (\frac{x}{2}) d x

\int x e^{3 x + 1} d x

y = 4 x^{2}, y = 4

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