Examples of interesting and creative problems about differentiation (in one variable) Problems about differentiation of functions in one variable that we find in the majority of textbooks are usually boring, that is, they are only a simple application of very known rules. So, what I want in this post is examples of derivatives (functions of one variable) that are interesting to take. I'd like that the problems were original, but if they aren't, feel free to share the same way. 
 A: Here are two examples that involve conic sections.

  
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*Suppose $a > 0$, and consider the line $L$ tangent to $y = \frac{1}{x}$ at $x = a$. Find the area of the triangular region between $L$, the $x$-axis, and the $y$-axis.
  

This problem's fun because it turns out the area doesn't depend on $a$ at all! I remember this is from Stewart's Calculus.


  
*Given the parabola $f(x) = a(x - h)^2 + k$ where $a > 0$, consider its tangent line at $x = x_0$. Show that this tangent line crosses the parabola's axis of symmetry $f(x_0) - k$ units below the vertex.
  

I'm having a hard time wording this one. But the cool fact is that if we focus on the $y$-coordinates only, $(x, f(x_0))$ and the intersection of the tangent line with the axis of symmetry will be equally spaced on either side of the vertex. I don't remember where I saw this one.
I guess the "theme" for both problems is to have to differentiate and work comfortably with parameters floating around. In the first, you can try a couple of values of $a$ to get a sense of what's happening, but you eventually have to work with $a$ as a fixed but unknown value. It's similar with the second, but more involved. Possibly so involved that you'd want to give a specific parabola if you don't think you have really strong students.
A: Here's an exercise about proving Young's inequality $$ab \le \dfrac1p a^p + \dfrac1q b^q \quad \forall a,b > 0,$$ where $\frac1p+\frac1q=1$ by finding the minimum of the function $f(x) =\frac1p x^p − x +\frac1q$ defined on $[0,+\infty)$ using the first and second derivatives of $f$ with respect to $x$.
