What is (\partial)/(\partial x)((yx)/(1+yx)) ?

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

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(\partial)/(\partial x)((yx)/(1+yx))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

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