I’m embarrassed to admit I’m getting my masters degree in a math related subject and I still don’t get this!
I know how to do them, but the way you compute them is almost suspiciously simple. What’s the logic behind converting the exponent to a constant? How does that determine the slope?
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One thing to recognize is that it is an exception to have a nice derivative. In general, the property of even having a derivative is quite rare (much more rare than having an integral), and having a *simple* derivative is even more rare. The only reason we can get the impression that derivatives are easy and simple is because we mostly just focus on functions with easy derivatives because they are easy. It’s a sample bias.
The reason the simple ones *are* simple is because they are constructed from the simple/basic functions of powers, trig functions, and exponents. Each of *these* have simple derivatives because they have very nice and simple addition formulas. For instance, the binomial theorem tells us what (x+y)^(n), the power of a sum is in terms of other powers. The formula e^(x+y)=e^(x)e^(y) tells us what the exponent of a sum is in terms of other exponents. The angle addition formula tells us what sine/cosine are of a sum of angles in terms of other sines/cosines. These formulas make the computation of the difference quotient manageable. The derivatives of functions which cannot be constructed from these nice-sum functions are much more difficult to work with. Luckily, we have power series and Fourier series which expand what we can construct with these nice functions, so even then it becomes computationally manageable. Which, honestly, is a miracle.
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