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 Differential Equations Ordinary and Partial Differential Equations Math Forum

 December 27th, 2015, 11:02 AM #1 Newbie   Joined: Dec 2015 From: PL Posts: 1 Thanks: 0 Derivative from definition Hello i have a little problem with this Deriverative from definition do i correctly arranged it? cause i have some problem with with the result it comes 0 on numerator and good value on denominator December 27th, 2015, 11:45 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra \begin{aligned} \lim _{x \to x_0}{{1 \over \sqrt x} - {1 \over \sqrt x_0} \over x- x_0} &= \lim_{h \to 0} {{1 \over \sqrt{x_0+h}}-{1 \over \sqrt x_0} \over h} \\ &= \lim_{h \to 0} \frac1h \cdot {\sqrt x_0 - \sqrt{x_0+h} \over \sqrt{x_0+h}\sqrt x_0} \\ &= \lim_{h \to 0} \frac1h \cdot {\sqrt x_0 - \sqrt{x_0+h} \over \sqrt{x_0+h}\sqrt x_0} \cdot {\sqrt x_0 + \sqrt{x_0+h} \over \sqrt x_0 + \sqrt{x_0+h} } \\ &= \lim_{h \to 0} \frac1h \cdot {x_0 - (x_0+h) \over \sqrt{x_0+h}\sqrt x_0 \left(\sqrt x_0 + \sqrt{x_0+h} \right) } \\ &= \lim_{h \to 0} \frac1h \cdot {-h \over \sqrt{x_0+h}\sqrt x_0 \left(\sqrt x_0 + \sqrt{x_0+h} \right) } \\ &= \lim_{h \to 0} {-1 \over \sqrt{x_0+h}\sqrt x_0 \left(\sqrt x_0 + \sqrt{x_0+h} \right) } \\ &= -{1 \over \sqrt x_0\sqrt x_0 \left(\sqrt x_0 + \sqrt x_0 \right) } \\ &= - {1 \over x_0 \cdot 2\sqrt x_0 } \\ &= -{1 \over 2 x_0^\frac32} \end{aligned} January 23rd, 2016, 06:25 AM   #3
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 Originally Posted by chrisplease Hello i have a little problem with this Deriverative from definition do i correctly arranged it? cause i have some problem with with the result it comes 0 on numerator and good value on denominator
I don't know what you mean by a "good value on denominator". If you take the limit of numerator and denominator separately, you should get 0 for both. Instead "rationalize the numerator" by multiplying both numerator and denominator by $\displaystyle \frac{\sqrt{x_0}+ \sqrt{x_0+ h}}{\sqrt{x_0}+ \sqrt{x_0+ h}}$ as v8archie did. (His "h" is $\displaystyle x- x_0$.) Tags definition, derivative, deriverative 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 Mathbound Calculus 2 September 20th, 2015 06:30 PM Scorks Calculus 3 May 5th, 2012 05:06 PM bewade123 Real Analysis 11 December 13th, 2011 05:07 PM salseroplayero Calculus 5 March 1st, 2010 11:44 AM mathman2 Calculus 2 October 19th, 2009 02:21 PM

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