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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{s}{( s ^{2} + 1 ) ( s ^{2} + 4 )}

x \to 1 lim (2 x^{3} - 5 x^{2} + x + 9)

\int \frac{3 x ^{2} + 2 x - 10}{x ^{3} + x ^{2} - 10 x + 8} d x

\int \frac{1}{x ln ( x )} d x

\int_{0}^{1} \frac{2 x + 3}{( x + 1 ) ^{2}} d x

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