structural:expansion
Example: arithmetic operations

If the function f is addition, that is, if

then D 1 f = ∂ f ∂ u = 1 {\textstyle D_{1}f={\frac {\partial f}{\partial u}}=1} and D 2 f = ∂ f ∂ v = 1 {\textstyle D_{2}f={\frac {\partial f}{\partial v}}=1}. Thus, the chain rule gives

For multiplication

the partials are D 1 f = v {\displaystyle D_{1}f=v} and D 2 f = u {\displaystyle D_{2}f=u}. Thus,

The case of exponentiation

is slightly more complicated, as

and, as u v = e v ln u, {\displaystyle u^{v}=e^{v\ln u},}

It follows that