Synthesis: Q and A A given function can be expressed as a simple expression in terms of variables xand y, but ultimately what is important are the variables u and v. So x and y aregiving you a nice formula for f, but actually the relevant variables for your problemare u and v. And you know how x and y are related to u and v. So of course what youcould do is plug the formulas the way that we did, substituting x with u and y with v,but maybe that will give you very complicated expressions. The important point here is that we don't need to know the actual formulas. All weneed to know is the rate of change. If we know all these rates of change, then weknow how to take these derivatives without actually having to plug in values. Yes,one could certainly do the same things in terms of x if y and z were functions of tinstead of being functions of u and v. Then it would be the same thing. You would have the same formulas that we had. So why does that one have straightd's? The answer is that if we want to put curly d's, we can do so but then we end upwith a function of a single variable. If you have a single variable, then the partial with respect to that variable is the samething as the usual derivative. Therefore, you do not need to worry about curlybrackets in this case. It is a special case of this one where instead of x and ydepending on two variables, u and v, they depend on a single variable, t. If you havea function of 30 variables, things work the same way just longer and you'll run out ofletters in the alphabet before the end. What would the effect on u and v be if one ofthem depended on another variable? If u and v themselves depend on another variable, then you would continue with yourchain rules. Maybe you would know how to express partial x partial u in terms usingthe chain rule. If u and v depend on yet another variable, then you could get thederivative with respect to that using first the chain rule to pass from u v to that newvariable. It is often easier to manipulate differentials than to use the product rule. So if youhave several substitutions to do, you can always arrange to use one chain rule at atime. You just have to do them in sequence. Is there a difference between df and curly df? The letter f does not exist. It is the variable of a function, and the only time that thevariable f appears alone is when it is equal to zero. A variable has no meaning byitself; it is always part of a function. The derivative df exists only when the other
variables are fixed. However, df represents only a partial rate of change with respectto f, leaving the other variables free to move.