Example 2.2.1: Gradient of a scalar function

We need the three partial derivatives. When differentiating with respect to one variable the other two are held constant, and the chain rule is used for the inner functions $xy$ and $yz$ (for example $\partial \cos (xy)/\partial x=-\sin (xy)\cdot \partial (xy)/\partial x=-y\sin (xy)$), while the factor $x$ of the second term is a constant for the derivatives with respect to $y$ and $z$:

Hence, by (2.2.2),

For instance at the point $(1,0,0)$ we get $\nabla f=a_x$: there $f$ increases fastest in the $+x$ direction, at the rate of one unit of $f$ per unit length.