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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> the way you compute them is almost suspiciously simple
They are designed that way. If you consider differentiation or integration as operators, OP say, they are both linear (f and g being some functions):
OP(a.f + b.g) = a.OP(f) + b.OP(g)
i. e. “transparent” to sums and multiplication by a scalar/constant. It can’t get much simpler than that.
The derivative, if it exists, embodies the idea that you can get information about a function *locally* (meaning near some point) if you approximate it there by something very simple, a linear function.
The integral, on the other hand, tries to synthesize information about a function’s behavior over a chunk of numbers (e. g. an interval). Up to some technical details it basically computes the *mean* of the function over the chunk. So it gets you a *global* information.
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