Background
The rules for differentiating elementary functions (arithmetic, exponential, trigonometric, etc.) together with the chain rule for differentiating compositions of functions are often considered the only basic inference tools needed to explicitly compute the derivatives of typical functions thrown at a beginning calculus student. Yet, an easy counterexample is $f(x) = x^x$, exploiting the fact that exponentiation is really a function of two variables. Then implicit / logarithmic differentiation techniques are typically introduced.
I had been curious in my calculus days about an explicit method to compute the derivative of $f$; what I came up with I haphazardly dubbed the independence trick: group all appearances of the independent variable into several partitions. Then, the derivative of the function is a sum, each term being a "partial" derivative of the function with respect to one partition, treating each instance outside of the partition as a constant. Rigorously, this is an easy consequence of the multivariate chain rule: $$ \left.\frac{d}{dt} f(x_1,\ldots,x_n)\right|_{(x_1,\ldots,x_n) = (x,\ldots,x)} = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(x,\ldots,x) \frac{\partial x_i}{\partial t}~~, $$ yielding the mentioned rule because each $\partial x_i / \partial t = 1$, as the variables are identical.
There are a few results nontrivial to a student without multivariate calculus:
- Directly, $\frac{d}{dx} x^x = x x^{x-1} + x^x \ln x = x^x (\ln x + 1)$.
- A one-line proof extending the Leibniz integral rule to its general form with variable limits.
Questions
Are there other nice applications for the "independence trick" in single-variable calculus?
Does anybody know if this method has been written / used / taught specifically as a trick for undergrads, and if so, where?