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

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

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

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

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

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

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

Simplify

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

\frac{\partial}{\partial x} (5 e^{x^{2} - 4 y})

\frac{d}{d x} (\frac{x ^{2} + 4 x - 5}{ln ( x )})

n = 1 \sum \infty n! e^{- n}

\frac{d y}{d x} = y^{2} sin (x^{2})

x \to - 1 - lim (\frac{3}{x ^{2} - 1})

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