Differentiate with respect to $y$: $f(x,y,z) = \sin(x)\sin(y)\sin(z)-\cos(x)\cos(y)\cos(z)$

Question

The derivative with respect to $y$ of

$$f(x,y,z) = \sin(x)\sin(y)\sin(z)-\cos(x)\cos(y)\cos(z)$$

is

a) $f’(x,y,z) = \cos(x)\cos(y)\sin(z) + \sin(x)\sin(y)\cos(z)$

b) $f’(x,y,z) = \sin(x)\cos(y)\sin(z) + \cos(x)\sin(y)\cos(z)$

c) $f’(x,y,z) = \cos(x)\cos(y)\cos(z) + \sin(x)\sin(y)\sin(z)$

d) $f’(x,y,z) = \sin(x)\sin(y)\sin(z) + \cos(x)\cos(y)\cos(z)$

Answer 1

If we differentiate with respect to $y$ the other variables are assumed to be constant, so we get $$\frac{\partial f(x,y,z)}{\partial y}=\cos(y)\sin(x)\sin(z)+\sin(y)\cos(x)\cos(z)$$