The time is approaching when you will want and need to find derivatives quickly. I am afraid that, with the exception of the product rule, I have no particularly clever ideas of how to how to teach this.
I am inclined to offer some explanation, short of a lot of proofs, to students as to why the rules and procedure are what they are. To that end I would start with some simple formulas using the (limit) definition of derivative.
- The derivative of constant times a function is the constant times the derivative of the function. This is easy enough to show from the definition since constants may be factors out of limits
- Likewise, the derivative of a sum or difference of functions is the sum of difference of the derivatives of the functions. This too follows easily from the properties of limits.
- For powers, keeping in mind the guesses from mention previously in the previous two posts “The Derivative I and II”, I suggest the method that all the books show. For example to find the derivative of x3 write
And perhaps a one or two more until the students are convinced of the pattern.
- For trigonometric functions follow your textbook: use the definition and the formula for the sine of the sum of two numbers along with the two special limits.
The next post will concern the product and quotient rules.
Teaching Calculus 
I’m curious–how important do you think the “special limits” are? Once students learn L’Hopital’s Rule, they will be able to do them. Do they really need to memorize separate rules for them?
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By “special limits” I assume you mean
and 
.
I think the importance of these is in developing the rule for differentiating sin(x). They are important for that. Certainly you should show them how that rule is developed, so you will need them. Since this is before L’Hopital’s Rule, you cannot use L’Hopital’s Rule.
Once you have developed L’Hopital’s Rule of course you can do without these two limits and any other limits that meet L’Hopital’s hypotheses. So they certainly do not need to be memorized for the exam.
In the bigger picture, I think that these limits, their own proof, and later their use are part of the beauty of mathematics and should be included for that reason.
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