December 27th, 2014, 07:37 AM  #1 
Senior Member Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36  Differentiation
Show that $\displaystyle y=\left x2 \right $ is not differentiable at $\displaystyle x=2$ How to show? 
December 27th, 2014, 08:33 AM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 3,031 Thanks: 1620 
derivative at a point $\displaystyle x = a$ is defined as $\displaystyle f'(a) = \lim_{x \to a} \frac{f(x)  f(a)}{xa}$ for $\displaystyle f(x) = x2$, show that the limit ... $\displaystyle \lim_{x \to 2} \frac{f(x)f(2)}{x2}$ ... does not exist. 
December 27th, 2014, 01:34 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,834 Thanks: 733  The limits for x > 2 and x < 2 are different.

December 27th, 2014, 06:01 PM  #4 
Senior Member Joined: Oct 2014 From: EU Posts: 224 Thanks: 26 Math Focus: Calculus 
You can consider the function as a piecewise function: $\displaystyle f(x) = \left \{ \begin{matrix} f_a(x) = x  2 &\text{if}\,x \ge 2 \\ f_b(x) = x+2 &\text{if}\,x < 2 \end{matrix}\right. $ Now, you can see that $\displaystyle f(x)$ is continous at $\displaystyle x=2$ evaluating the limits for $\displaystyle f_a(x)$ and $\displaystyle f_b(x)$ when $\displaystyle x$ tends to 2, but each derivative is different: 1 and 1, so $\displaystyle f(x)$ is not differentiable when $\displaystyle x = 2$. Last edited by szz; December 27th, 2014 at 06:21 PM. 

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