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 December 12th, 2015, 02:19 PM #1 Member   Joined: May 2015 From: Earth Posts: 64 Thanks: 0 Limit rules trick question? Use the definition of the derivative (and limit rules) to find the derivative of |x^3| at x = 0 Is this a trick question? Like just from looking at it, I know the derivative of an absolute value such as x^3 does not equate at x=0, right? December 12th, 2015, 02:58 PM #2 Math Team   Joined: Jul 2011 From: Texas Posts: 3,002 Thanks: 1587 $\displaystyle f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ $\displaystyle f'(0) = \lim_{x \to 0} \frac{|x^3| - 0}{x - 0}$ $x < 0 \implies |x^3| = -x^3$ $\displaystyle \lim_{x \to 0^-} \frac{-x^3}{x}$ $\displaystyle \lim_{x \to 0^-} -x^2 = 0$ $x \ge 0 \implies |x^3| = x^3$ $\displaystyle \lim_{x \to 0^+} \frac{x^3}{x}$ $\displaystyle \lim_{x \to 0^+} x^2 = 0$ limit from the left = limit from the right, therefore ... $\displaystyle f'(0) = \lim_{x \to 0} \frac{|x^3| - 0}{x - 0} = 0$ December 12th, 2015, 04:53 PM #3 Member   Joined: May 2015 From: Earth Posts: 64 Thanks: 0 So you're saying apply the sandwich rule or whatever? To take the derivative on both sides to get the limit? Upon looking at the graph I see there is no edge there like I thought there was. If there was an edge, there would be no limit there, or derivative? How does that all I work? I forget. Thanks. December 12th, 2015, 06:23 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,926 Thanks: 2205 It's straightforward to show from first principles that ax³ has derivative 3ax², with value 3a*0 = 0 at x = 0. Letting a = sgn(x) doesn't affect this result. That wouldn't work for |x|, as the derivatives of x and -x take different values at x = 0. The derivative of |x³| is 3x|x|, as can be verified by differentiating (x²)$^{3/2}$ using the usual rules. Tags limit, question, rules, trick Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post MetallicaBand Elementary Math 3 June 10th, 2014 10:54 AM Shamieh Algebra 3 May 29th, 2013 06:39 AM mathkid Algebra 3 February 25th, 2013 03:51 AM mathkid Calculus 1 September 22nd, 2012 11:42 AM tomorrow Calculus 14 September 22nd, 2012 03:05 AM

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