What is (\partial)/(\partial y)(-ysin(xy)) ?

The answer to (\partial)/(\partial y)(-ysin(xy)) is -sin(yx)-yxcos(yx)

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(\partial)/(\partial y)(-ysin(xy))

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

- sin (y x) - y x cos (y x)

Solution steps

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

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

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

f = y, g = sin (x y)

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

\frac{\partial}{\partial y} (y) = 1

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

= - (1 \cdot sin (x y) + cos (x y) x y)

Simplify

- (1 \cdot sin (x y) + cos (x y) x y) : - sin (y x) - y x cos (y x)

= - sin (y x) - y x cos (y x)

What is (\partial)/(\partial y)(-ysin(xy)) ?
The answer to (\partial)/(\partial y)(-ysin(xy)) is -sin(yx)-yxcos(yx)