What is (\partial)/(\partial y)(xyln(y)) ?

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

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(\partial)/(\partial y)(xyln(y))

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

x (ln (y) + 1)

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = y, g = ln (y)

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

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

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

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

Simplify

x (1 \cdot ln (y) + \frac{1}{y} y) : x (ln (y) + 1)

= x (ln (y) + 1)

x \to 0 lim (\frac{5 x}{x})

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