# Are there cases where the difference between a partial and total derivative is purely reliant on syntax?

It's always seemed to me that the difference between a partial and total derivative is artificial. I'm not sure how I'm wrong but surely, I must be. An example of my confusion is the function f(x,t)=x^2+t where x(t)=t. If I take the partial derivative of f w.r.t t, $\frac{\partial f}{\partial t}$=1. Now suppose I take the partial again, but this time I substitute first. f(x(t),t)=t^2+t. This time $\frac{\partial f}{\partial t}$=2t+1. This confuses me because, surely the result of a partial derivative relies only on the function f, and can't be dependent on my choice of how to write it. Unless the partial derivative is really such a whimsical, subjective operator.

• For starters, $f(x,t) = x^2 + t$ isn't a function if $x(t) = t$. $f$ would only be defined for pairs $(x,t)$ such that $x = t$, and would be undefined at points like $(0,1)$ or $(5,-3)$. Commented Sep 6, 2013 at 20:46
• Its domain is very limited, but I believe it's still a function. For example, 1/x is also undefined for some value of x, but it has a derivative. Regardless, I don't mean for this f(x,t) to be a specific confusion, rather it's just an example of why I'm confused. Commented Sep 6, 2013 at 20:57
• @Goos, it is a function of two variables. The thing is, composing it with $t\mapsto (t^2,t)$ creates a whole different function (of one variable), so naturally the first's partial w.r.t $t$ doesn't equal the second's derivative. Commented Sep 6, 2013 at 21:14

You have overloaded a variable for two uses: $t$ is used as the second slot of $f$ as well as a variable of the expression $f(x(t),t)$, so it causes confusion.
Suppose we use $f(x(u),u)$ instead. It wouldn't make sense to write $\partial f / \partial u$ here right? When you write partial derivative syntax, you have a particular directional derivative in mind (with respect to whichever variable slot you are curious about).
Now, suppose you also wrote $g(u) = f(x(u),u)$. It does make sense to write $\partial g / \partial u$ and $dg/du$. But it's silly because the partial derivative of $g$ in the direction of its only variable slot ($u$) gives total information about the derivative.