Finding First Derivatives by DefinitionDate: 10/30/98 at 22:13:53 From: Lisa Subject: Finding Derivatives by Definition I can't seem to understand how to find the first derivative by definition. The part of the equation I can't figure what f is in f(x-h) - f(x). Could you please show me a good sample problem? I can do it the easy way, but our teacher says we have to know how to do it by definition on our tests. Date: 10/31/98 at 03:19:34 From: Doctor Pat Subject: Re: Finding Derivatives by Definition Lisa, Here is a step-by-step approach that makes it manageable (NOTHING makes it easy.) I'll use the very easy f(x) = x^2 + 2x for my example. 1) Write out f(x) and f(x+h) to the side and simplify both: f(x) = x^2 + 2x f(x+h) = (x+h)^2 + 2(x+h) = x^2 + 2xh + h^2 + 2x + 2h Note (x+h) replaces x everywhere it appeared. 2) Now write out the Newtonian quotient with f(x+h) and f(x) as above: f(x+h) - f(x) [x^2+2xh+h^2+2x+2h] -[x^2+2x] ------------- = ----------------------------- h h 3) Simplify numerator as much as possible. Note that you should try to find an h in every term: 2xh + h^2 + 2h -------------- h 4) Now divide h in the numerator and denominator: 2x + h + 2 ---------- 1 5) Finally, evaluate the limit as h goes to zero: Lim 2x + h + 2 = 2x + 0 + 2 = 2x + 2 h->0 Sometimes the simplification in steps two and three takes much more work, but this is a useful pattern to work from. Good luck. - Doctor Pat, The Math Forum http://mathforum.org/dr.math/ |
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