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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

(e^{x} - 1)^{^{'}}

x e^{\frac{x}{2}}

d cos (t) + t^{2} sin (t)

n \to \infty lim (\frac{6}{n})

\int \frac{3}{7 + 3 x} d x

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