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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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