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

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

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

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

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

Solution steps

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

Treat

z, y

as constants

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

x^{^{'}} = - 2 x

x \to - 10 lim (- (\frac{1}{x + 10}))

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

\frac{d}{d x} (a x^{2} e^{3 x})

\int 2 cos (x) cos (2 x) d x

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