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

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

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

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

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

Solution steps

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

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

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

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

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

Simplify

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

\frac{d}{d x} ((sin (9 x))^{l n (x)})

\int 10 e^{5 x} d x

\frac{d}{d x} (tan (cos^{2} (x^{5})))

x \to 0 lim (\frac{7 x ^{2} - 4 x}{x})

\frac{d}{d t} (a sin (t))

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