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

The answer to (\partial)/(\partial y)(xln(1+x+y)) is x/(1+x+y)

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(\partial)/(\partial y)(xln(1+x+y))

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

\frac{x}{1 + x + y}

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

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

= x \frac{1}{1 + x + y} \cdot 1

Simplify

x \frac{1}{1 + x + y} \cdot 1 : \frac{x}{1 + x + y}

= \frac{x}{1 + x + y}

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What is (\partial)/(\partial y)(xln(1+x+y)) ?
The answer to (\partial)/(\partial y)(xln(1+x+y)) is x/(1+x+y)