A partial derivatives question

Does the product rule from normal differentiation apply also with partial differentiation? Im not quite sure if I got this right.

I need to find the partial derivative δz/δy of z = -xysin(y)

What I've done is split it into 2 parts,

-xy and siny

Then I used the product rule, but only including 'y' and making 'x' a constant

So following the formula:

f'g+fg'

f =-xy
f' = -x
g = sin(y)
g' = ycos(y)

so I get δz/δy = -xsiny-x(y^2)cos(y)

Is this correct? If not, what is the right way to approach it?

You've got the right idea. Just a small error in your differentiation...

δz/δy of z = -xysin(y)

f = -xy
f' = -x
g = sin(y)
g' = cos(y) <== not y*cos(y) because the angle is what you are differentiating with respect to.

δz/δy = -xy * cos(y) - x * sin(y)

Simplified:

δz/δy = -x [y * cos(y) + sin(y)]

Everything is good, except g' = cos(y)

∂z/∂y = -x{sin(y)} - x(y){cos(y)}

z = (-xy) (sin y)

∂z/∂y = (-x) sin y + cos y (-xy)

∂z/∂y = - x sin y - (xy) cos y