What is (\partial)/(\partial y)(e^{xy}ln(y)) ?

The answer to (\partial)/(\partial y)(e^{xy}ln(y)) is e^{xy}xln(y)+(e^{xy})/y

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(\partial)/(\partial y)(e^{xy}ln(y))

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

e^{x y} x ln (y) + \frac{e ^{x y}}{y}

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

f = e^{x y}, g = ln (y)

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

\frac{\partial}{\partial y} (e^{x y}) = e^{x y} x

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

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

\frac{1}{y} e^{x y} = \frac{e ^{x y}}{y}

= e^{x y} x ln (y) + \frac{e ^{x y}}{y}

What is (\partial)/(\partial y)(e^{xy}ln(y)) ?
The answer to (\partial)/(\partial y)(e^{xy}ln(y)) is e^{xy}xln(y)+(e^{xy})/y