Sarahisme Posted April 30, 2005 Share Posted April 30, 2005 how do you show/prove that a function is differentiable at EVERY real x?? lol i just cant figure out the method? i thought of induction but yeah....guess not anyways yep that it Sarah Link to comment Share on other sites More sharing options...
alext87 Posted April 30, 2005 Share Posted April 30, 2005 every function at every point except at infinity has a gradient therefore every function is differentiable as long as you can prove it does go to infinity - is this statement correct? Link to comment Share on other sites More sharing options...
Johnny5 Posted April 30, 2005 Share Posted April 30, 2005 how do you show/prove that a function is differentiable at EVERY real x?? lol i just cant figure out the method? i thought of induction but yeah....guess not anyways yep that it Sarah you might want to check this out: Wolfram on differentiable functions Regards Link to comment Share on other sites More sharing options...
matt grime Posted April 30, 2005 Share Posted April 30, 2005 It would depend on te function. sqrt(x) on the positive reals (ie including zero) is not differentiable. On the strictly positives it is differentiabel at all points. You simply show that for any given point of the domain the function is differentiable. Unless you have specific example in mind there is nothing more that anyone can say to you. alext87's post is absolutely wrong. Link to comment Share on other sites More sharing options...
Ducky Havok Posted May 1, 2005 Share Posted May 1, 2005 Like matt grime said, it depends on the function. You can check a few things though. If you want it to be differentiable at every real x then it has to be continous at every real x. Also, make sure there are no sharp points, like in the absolute value function at 0. Link to comment Share on other sites More sharing options...
Sarahisme Posted May 1, 2005 Author Share Posted May 1, 2005 well the function is |x|^3 btw Ducky, your picture is awesome! Link to comment Share on other sites More sharing options...
Sarahisme Posted May 1, 2005 Author Share Posted May 1, 2005 ok ok the question is how do you use the defintion of dervative (f'(x) = (f(x+h - f(x))/h) to calucalte the dervative of |x| or |x|^3 ?? Link to comment Share on other sites More sharing options...
Sarahisme Posted May 1, 2005 Author Share Posted May 1, 2005 don't worry i think i got it! Link to comment Share on other sites More sharing options...
matt grime Posted May 2, 2005 Share Posted May 2, 2005 or you could use the fact that |x|^3 is just x^3 if x is positive, and -x^3 if x is negative, and thus clearly differentiable at all points but 0. so it suffices to check the derivative at 0, from the right you get the derivative being as for x^3, and the left as -x^3, both of which give 0 at the origin, so it is differentiable. Link to comment Share on other sites More sharing options...
Guest eKo.RavanDi Posted May 3, 2005 Share Posted May 3, 2005 first of all the function should be continous at all of the points in its domain. because there is no derivation in the end-points. then you should check for the sharp points. each point is NOT a sharp point if the left and right derivatives are equal. f'+(a)=f'-(a). the left and right derivatives are the left and right limits of f(x)-f(a)/x-a when x gets near to a. sorry for bad english, Link to comment Share on other sites More sharing options...
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