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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

x \to 7 lim (x^{2} - 49)

\int \frac{200}{1 + x} d x

\frac{1 + 5 s ^{2}}{s ^{2} ( s - 9 )}

\frac{3}{2 s}

\int x^{2} - x + 1 d x

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