Differentiation Help: Solving x|x| & g(x) Problems

[semidevil]

i'm looking at this, but don't really know how to approach this.

determine where the function x|x| is differentiable and find the derivative.

i'm looking at this, but just have no idea where to start? I mean, I know x is differentiable everywhere, and |x| is not differentable at 0. looking at this, I'm thinking about the product rule, and I'm tryint to see if I need to solve this using epsilon delta method, but have no idea.

any clues just on how to solve these types of problems?

and another one:

g(x) = x^2/xin(1/x^2) for x != 0 and g(0) = 0. I'm suppose to show that g is differntaible everywhere.

how do I show these types of problems??

 

[mathman]

For x not=0, there is no problem with either of your examples. To handle the case of x=0, you could try forming difference quotients and letting the increment go to 0. For both examples you have to work with two difference quotients, positive increment and negative increment. If they both have the same limit as the increment goes to 0, then the derivative exists.

 

[Jameson]

Another way is to analyze the function graphically. If there are any discontinuities or "sharp turns" where the limits on either side are not the same, then the function is not differentiable there.