MHB General question, which formula to use to find derivative?

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SUMMARY

This discussion addresses the selection of formulas for finding derivatives in Calculus 1. Two primary formulas are highlighted: the limit definition of the derivative using the Binomial Theorem and the limit definition involving factoring. The example provided illustrates that for the function $f(x)=x^n$, the choice of formula depends on the student's comfort with either expanding $(x+h)^n$ or factoring $y^n-x^n$. Ultimately, students are encouraged to experiment with both methods to determine which is more manageable for them.

PREREQUISITES
  • Understanding of basic calculus concepts, specifically derivatives.
  • Familiarity with the Binomial Theorem.
  • Knowledge of limit definitions in calculus.
  • Ability to factor polynomial expressions.
NEXT STEPS
  • Study the Binomial Theorem in detail to enhance derivative calculations.
  • Practice using the limit definition of derivatives with various polynomial functions.
  • Explore factoring techniques for polynomials to simplify derivative problems.
  • Review examples of both derivative formulas to identify strengths and weaknesses in their application.
USEFUL FOR

Students in Calculus 1, educators teaching introductory calculus, and anyone seeking to improve their understanding of derivative calculations and methods.

theothersophie
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Hi this is my first post ever in MHB, and I'm in Calculus 1 wondering which formula to use to find derivatives. There are 2 as far as I know:

(1)
derivative_definition_formula.gif


and the one one at the beginning of this:

(2)
calc2-1notes14.gif


Example Problem:

LFH5oUF.png


How would i know which formula to use? Is there a particular reason the example used (2)?

Thanks all. This is a quick post so I'm sorry if I'm unaware of the rules here, you can point me in the right direction to unerstand formatting and rules around MHB.
 
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Experience is the only teacher as to which formula to use. For example, if $f(x)=x^n$, then using
$$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$
requires you to use the Binomial Theorem to expand out $(x+h)^n$ satisfactorily. On the other hand, if you use
$$f'(x)=\lim_{y\to x}\frac{f(y)-f(x)}{y-x},$$
then you must factor $y^n-x^n$ to get the factor $y-x$ to cancel. Different students might find the one or the other more difficult or harder to implement. Bottom line: try both, and see which one works out easier.
 

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