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

The answer to (\partial)/(\partial x)(e^{xy}cos(x+y)) is e^{xy}ycos(x+y)-e^{xy}sin(x+y)

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

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

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

Solution steps

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

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 + y)

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

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

\frac{\partial}{\partial x} (cos (x + y)) = - sin (x + y)

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

Simplify

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

\frac{d}{d x} (b x e^{x})

\frac{\partial}{\partial x} (\frac{2 x ^{2} y}{5 + x ^{2} y ^{2}})

\frac{d}{d x} (((x^{2} + 3)^{5} + x)^{2})

ln (5 + x)

\frac{\partial}{\partial x} (8 y^{5} + 8 y^{2} x^{7})

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