What is (\partial)/(\partial x)(e^{xy}sin(xyz)) ?

The answer to (\partial)/(\partial x)(e^{xy}sin(xyz)) is e^{xy}ysin(xyz)+cos(xyz)yze^{xy}

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(\partial)/(\partial x)(e^{xy}sin(xyz))

\frac{\partial}{\partial x} (e^{x y} sin (x yz))

e^{x y} y sin (x yz) + cos (x yz) yz e^{x y}

Solution steps

\frac{\partial}{\partial x} (e^{x y} sin (x yz))

Treat

y, z

as constants

Apply the Product Rule:

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

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

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

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

\frac{\partial}{\partial x} (sin (x yz)) = cos (x yz) yz

= e^{x y} y sin (x yz) + cos (x yz) yz e^{x y}

What is (\partial)/(\partial x)(e^{xy}sin(xyz)) ?
The answer to (\partial)/(\partial x)(e^{xy}sin(xyz)) is e^{xy}ysin(xyz)+cos(xyz)yze^{xy}