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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

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

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

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

Simplify

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

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

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