# Thread: Determining if a function is differentiable

1. ## Determining if a function is differentiable

I can't figure out how to do this at all. Basically...the GRAPH looks differentiable, and I know from my book that $\displaystyle g(x) = (x + |x|)^2 + 1$ is differentiable, so...shouldn't this one be too? According to my math (taking the limit of both sides)...it's not. I get the left hand limit is -1 and the right hand limit is 0. And I don't get why, or if that's right...?

Problem: Determine whether or not the function $\displaystyle g(x) = (x + |x|)^2 - 1$ is differentiable at x = 0. If so, find $\displaystyle g\prime(0)$.

Also, to do it, I can only use a difference quotient (I haven't learned any fancy shortcuts yet):

$\displaystyle f\prime(x) = \frac{f(x + h) - f(x)}{h}$

2. to check differentiability, the following limit must exist $\displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac{g(x)-g(0)}{x-0},$ and not only that, check the one-sided limits, they must exist and be equal each other.

3. Originally Posted by Krizalid
to check differentiability, the following limit must exist $\displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac{g(x)-g(0)}{x-0},$ and not only that, check the one-sided limits, they must exist and be equal each other.
Also you might notice that:$\displaystyle g(x) = \left\{ {\begin{array}{rl} {4x^2 - 1,} & {0 \leqslant x} \\ { - 1,} & {x < 0} \\ \end{array} } \right.$