Examples of a problem solved by a well-chosen derivative equaling zero What are examples of problems which are solved by taking a derivative of a well-chosen function, and finding that it is zero, therefore the function must be constant?
I can think of a few: 
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*Show $\int_0^pf(t)dt = \int_a^{a+p}f(t)dt$ for $f$ periodic of period $p$. This can be solved by differentiating $\int_0^pf(t)dt - \int_a^{a+p}f(t)dt$ with respect to $a$, noting that it is zero, and noting that the function has value zero at $a=0$. 

* Show that $\int_1^a \frac1t dt = \int_b^ab \frac1t dt$. This can be shown by differentiating $\int_1^a \frac1t dt - \int_b^ab \frac1t dt$ with respect to $b$, noting that it is zero, and noting that the function is zero ab $b=1$.


*
A bounded harmonic function on $\mathbb{R}^n$ must be constant, since we can show by mean value property that $|\frac{\partial u}{\partial x_i}| \leq C/r \cdot \max_{\|x\| =r}|u|$ for any $r$, so sending $r$ to infinity shows $u$ is a constant.


I note that both of these first two problems are similar in that what is of interest is in the limits of an integral. I was looking for some other examples, perhaps not sharing this similarity. I'm sure there are far too many examples to list, but who has some good ones?

In elementary calculus, one  shows that the solutions of   $\frac{dy}{dt}=ky$ are $y=Ce^{kt}$ by letting $w=ye^{-kt}$. Then $\frac{dw}{dt}=0$.  
