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

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

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(\partial)/(\partial x)((x+y)/(1-xy))

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

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

Solution steps

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

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

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

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

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

Simplify

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

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

f (x) = x^{3} - 3 x

(x^{2} - 5) (x + 7)

\int sin (11 x) d x

\int_{0}^{2} x^{2} - 4 d x

\frac{\partial}{\partial y} (3 z)

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