Should Differential or Integral Calculus be Taught First?

[The_Z_Factor]

Does it really matter which calculus is taught first? A book I had that I've recently returned to the library was written by a mathematician who said that neither of the subjects are harder but integral is usually taught after differential. He also said that sometimes people are taught integral before differential. My current teacher (not exactly a teacher more like a person who assists me when I have trouble, since technically I am not in school) told me he was taught differential first and then integral.

So it seems that integral would be a bit more difficult or perhaps complex than differential calculus? The only reason I could find for teaching/learning differential before integral is that when I was looking in my calculus book the other day I skipped a lot of chapters and went to the integral part. My book says its basically backwards differentiating, and it uses some laws that I learned when I started on differential calculus like the power rule etc.

So would it really affect you significantly if you learn integral before differential or vice versa? Because if integrating is backwards differentiating, then differentiating is backwards integrating, seems to me like it wouldn't really matter that much.

 

[Ben Niehoff]

The difference is that, for differentiation, there are set procedures we can follow to systematically find the derivative of any differentiable function. But for antiderivatives, no general procedure exists; really, the only way to get antiderivatives is to remember a bunch of common derivatives and recognize them when they appear under the integral sign.

If you attempted to teach integral calculus first, there are a few common antiderivative formulae you could give, but there is not a straightforward way to derive them---contrast to the derivative formulae, which can always be derived from

f'(x) = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}

I suppose, if pressed, you could try to get antiderivative formula from the limit of Riemann sums, but, well, good luck...

 

[neutrino]

One famous text that teaches integrals (definite integrals, specifically) first is Apostol's Calculus Vol.1. Differentiation/derivatives is/are not mentioned during the development of the definite integral from a few axioms on area.

But solving integrals is different. If you want to solve them fast, you will have to remember a few formulae and practice a lot.