How much do you memorize to do general derivatives/integrals?

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How much do you memorize to do general derivatives/integrals? While it is possible to do everything by going back to first principles, I imagine it gets exhausting to do so every time. So which derivatives/integrals do you memorize? Are the ones that I have attached to this post good enough to have memorized?
 

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  • #2
Stephen Tashi
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How much do you memorize to do general derivatives/integrals?
You aren't describing a specific goal. Are you asking how much a student in second semester calculus should memorize? He would also need to know something about exponential functions.
 
  • #3
arildno
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Well, I haven't thought of it, here's my typical rule
1. monomials of x (if power -1, logarithm)
1b). If in fractions, use fractional decomposition, logaritms for linear factors, arctan for irreducible squares.
2. Qaudratics under root sign: use trig subs,
3. Rational functions of polynomials in trig functions: Tan(pi/2)-substitutions
4. And exponentials.
5. Fiddling with product rule and substitution rule to get from 1 to 4, or give up.
That's about it, I think.
 
  • #4
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You aren't describing a specific goal. Are you asking how much a student in second semester calculus should memorize? He would also need to know something about exponential functions.
I finished Calculus 1 and 2 and I'm currently taking a course in Differential Equations. I memorized the attachment above for Calculus 2 as advised by the textbook but I find that I have forgotten them over the break. It seems that I still need to have them memorized in order to do integration in Differential Equations but I want to get a general feel of how much I should re-memorize because I don't want to memorize more than I need to.
 
  • #5
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Well, I haven't thought of it, here's my typical rule
1. monomials of x (if power -1, logarithm)
1b). If in fractions, use fractional decomposition, logaritms for linear factors, arctan for irreducible squares.
2. Qaudratics under root sign: use trig subs,
3. Rational functions of polynomials in trig functions: Tan(pi/2)-substitutions
4. And exponentials.
5. Fiddling with product rule and substitution rule to get from 1 to 4, or give up.
That's about it, I think.
I see that you have a general algorithm for integration, but do you memorize any derivatives or integrals? For example: you probably have memorized that the derivative of sinx is cosx or that that the derivative of lnx is 1/x.
 
  • #6
mathwonk
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basically all you need to know to do derivatives is x' = 1, linearity, and the product and chain rules and the fundamental theorem of calculus. all the derivatives of all the exponential and log and sin and cosine functions and rational functions follow from those. of course you should really memorize the basic ones and not derive them every time.

antiderivatives are harder but again the main tools are linearity, the product rule (integration by parts) and the chain rule (integration by substitution).

To simplify some special integrals it helps to remember your trig identities.

however since only a few special integrals occur as derivatives of familiar functions, antidifferentiation is often of no use in real applied problems, so it is very important to also know how to approximate integrals using monotonicity, and to use power series.
 

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