What is (\partial)/(\partial x)(e^{xy}cos(x^2)) ?

The answer to (\partial)/(\partial x)(e^{xy}cos(x^2)) is e^{xy}ycos(x^2)-2e^{xy}xsin(x^2)

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(\partial)/(\partial x)(e^{xy}cos(x^2))

\frac{\partial}{\partial x} (e^{x y} cos (x^{2}))

e^{x y} y cos (x^{2}) - 2 e^{x y} x sin (x^{2})

Solution steps

\frac{\partial}{\partial x} (e^{x y} cos (x^{2}))

Treat

y

as a constant

Apply the Product Rule:

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

f = e^{x y}, g = cos (x^{2})

= \frac{\partial}{\partial x} (e^{x y}) cos (x^{2}) + \frac{\partial}{\partial x} (cos (x^{2})) e^{x y}

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

\frac{\partial}{\partial x} (cos (x^{2})) = - sin (x^{2}) \cdot 2 x

= e^{x y} y cos (x^{2}) + (- sin (x^{2}) \cdot 2 x) e^{x y}

Simplify

= e^{x y} y cos (x^{2}) - 2 e^{x y} x sin (x^{2})

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What is (\partial)/(\partial x)(e^{xy}cos(x^2)) ?
The answer to (\partial)/(\partial x)(e^{xy}cos(x^2)) is e^{xy}ycos(x^2)-2e^{xy}xsin(x^2)