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April 16th, 2011, 08:52 AM  #1 
Joined: Sep 2009 Posts: 106 Thanks: 0  how do you know if a func is Differentiable?
The solution in my book starts out by stating that this func is Differentiable. How do you know it is Differentiable? Let's say I'm asked to find all x values when this function is growing. I know that if a function is Differentiable then, it's growing, when it's derivative function values are bigger than zero. now solve for x & we get 2 x values: 1/3 & 3 Since 3x^2 is positive, we make the graph of this derivative function, starting from top right: From the graph we see that the derivative function values are bigger than zero when x is less than 1/3 and bigger than 3: so our answer is: this function is growing when x values are between [infinity :1/3] & [3;infinity] Second question is: Now let's suppose I do not know at first glance if a function is Differentiable or not, how can I find out when its growing? 
April 16th, 2011, 09:31 AM  #2  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 3  Re: how do you know if a func is Differentiable? Quote:
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And generally, in this context, functions are going to be continuous EVERYWHERE, except at a point (or a few points). Take, for instance, the absolute value function y = x. This function is NOT differentiable at x = 0, since the limit that is the definition of the derivative does not exist. But everywhere else (besides at x = 0), this function is differentiable. Functions are not differentiable where they are discontinuous, so that might be another good starting point. If you are asked to find where a function is increasing (i.e. "growing"), you would still take the derivative (whether or not it is differentiable everywhere!). Then just keep in mind the xvalues where the function is not differentiable, and set the derivative equal to zero to find the horizontal tangents. etc.  
April 16th, 2011, 09:32 PM  #3  
Joined: Apr 2007 Posts: 2,141 Thanks: 0  Quote:
For x = ?, the function has relative extrema. For ? < x < 3, the function is decreasing. For x = 3, the function has relative extrema. For x > 3, the function is increasing.  
April 16th, 2011, 10:43 PM  #4 
Joined: Apr 2011 Posts: 9 Thanks: 0  Re: how do you know if a func is Differentiable?
Simple, if a function is continuous i.e. it exists for a<x<b then it's differentiable. Because differentiating means finding the gradient of a function at a specific point. All functions have gradients! 
April 17th, 2011, 02:57 AM  #5  
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 11,808 Thanks: 267 Math Focus: The calculus  Re: how do you know if a func is Differentiable? Quote:
 
April 17th, 2011, 05:22 PM  #6  
Global Moderator Joined: Dec 2006 Posts: 11,318 Thanks: 271  Quote:
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April 17th, 2011, 05:59 PM  #7  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 3  Re: Quote:
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I know.  
April 17th, 2011, 06:14 PM  #8  
Global Moderator Joined: Dec 2006 Posts: 11,318 Thanks: 271  Quote:
 
April 17th, 2011, 06:32 PM  #9 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 3  Re: how do you know if a func is Differentiable?
Case 1. 
April 17th, 2011, 07:16 PM  #10 
Global Moderator Joined: Dec 2006 Posts: 11,318 Thanks: 271 
Which is what, and in reply to what?


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