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

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

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

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

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

Solution steps

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

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

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

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

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

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

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

\frac{\partial}{\partial x} (x^{0.5})

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

\int \frac{6}{x ^{3}} + 16 x^{3} d x

\frac{\partial}{\partial t} (e^{t} cos (t))

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