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

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

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

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

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

Solution steps

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

Simplify

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

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

= - 2 \cdot \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

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

Simplify

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

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

x \to 0 lim (\frac{x - 1}{x ^{2} - 1})

(x^{2} + 1) (x + 5 + \frac{1}{x})

\frac{\partial}{\partial y} (4 x e^{x^{2} + y^{2}})

y^{^{'}} = (3 x - 2)^{2} (30 x + 29)

t y^{^{'}} + (t + 1) y = 2 t e^{- t}

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