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

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

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

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

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

Solution steps

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

Treat

y, z

as constants

Take the constant out:

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

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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