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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Like others have said, this specific rule is just one that’s notoriously simple. Why it happens is a matter of combinatorics more than anything else. Look at Pascal’s triangle and then a definition of the derivative and you’ll pretty much see why it works out that way.
However, it’s not always that simple, especially for derivatives. Could you, off the bat, tell me what the derivative of arcsin(x) is? That’s still a relatively “simple” derivative but it is almost unrelated to the simple power rule you talked about. There’s really no rhyme or reason that a derivative has a particular form or pattern other than the math just checking out that way.
Integrals are even worse. Integrals are by and large far less simple to compute. There are some seemingly simple integrals that are literally impossible to express in terms of any other normal function..
Integral of x^2 ? Easy, (1/3)x^3 + c
Integral of cos(x^2 )? Good luck, go throw it into wolfram alpha and let me know what you get. It’s not pretty. But that function is so simple, right?
In short, there are certain rules that are known simply because they form patterns. There are no truly deep mathematicalreasons that any derivative/Integral rules work the way they do.
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