<p>But really, the repetition of proof just becomes a mechanism for memorizing the proof… :)</p>
<p>What you are really doing when taking a derivative is finding the function that models the rate of change of the given function. The function that models the rate of change is really based upon the collection of tangent lines that model these slopes. If you understand the basic concept of what a derivative is conceptually – for derivatives in general – then you don’t need to take that same idea and beat it upside the head with each and every single particular function to demonstrate that understanding.</p>
<p>I think both too much and too little stock is put into memorization:</p>
<p>Too much stock is put into memorization rather than learning the conceptual ideas behind the topic, especially with ideas that can easily be looked up when needed. It’s much more important to know when you need the derivative of sec x and be able to look up that said derivative is sec x tan x, then to not know when you need it in the first place.</p>
<p>But at the same time, too little stock is put into memorization when we go to such extremes to avoid said memorization. On the AP Test, knowing these derivatives cold will greatly increase your chances of getting a 5, if for no other reason than that you can knock out those trig derivative questions more quickly, leaving you more time to answer other questions that you might not have otherwise gotten to.</p>
<hr>
<p>As a side note, I would venture that it takes more than just a few minutes to adequately prove each identity.</p>
<p>For instance, the proof that d/dx[sin x] = cos x is fairly complicated:</p>
<p>dy/dx
= lim (h->0) [sin (x+h) - sin x]/h
= lim (h->0) [sin x cos h + cos x sin h - sin x]/h
= lim (h->0) [sin x (cos h - 1) + cos x sin h] / h
= lim (h->0) [sin x (cos h - 1)]/h + lim (h->0) [cos x sin h]/h
= sin x * lim (h->0) [cos h - 1]/h + cos x * lim (h->0) [sin h]/h</p>
<p>Where it turns out that lim (h->0) [cos h - 1]/h = 0 and lim (h->0) [sin h]/h = 1.</p>
<p>But how do you know those? Memorize them? Evaluate for numerous values of h that are progressively closer to 0 from both sides?</p>
<p>I would venture that short of either one of these techniques that you’d have difficulty proving these statements. The idea, of course, being that you can’t get away without memorizing anything.</p>
<p>Honestly, while it’s nice to be able to know the proofs of the derivatives of functions like that, it’s not the major point of AP Calculus AB/BC. The point is that you can apply your knowledge of derivatives to be able to answer what’s at the heart of calculus: the ideas of motion and change.</p>
<p>And someone who understands conceptually what a derivative is – in the particular case of y = sin x – can just as easily graph Y1 = nDeriv(sin x, x, x), notice that the result is the graph of cos x, and move on in the instance where they forget the result, which ultimately, I think will prove more meaningful than reproving it using the limit definition.</p>