# Thread: Differentiability

1. ## Differentiability

Find all values of x for which the function is differentiable.

$P(x)= sin(|x|)-1$

2. Originally Posted by Truthbetold
Find all values of x for which the function is differentiable.

$P(x)= sin(|x|)-1$
Can you think of any points of the graph of $y = |x|$ where the function isn't differentiable?

-Dan

3. Originally Posted by topsquark
Can you think of any points of the graph of $y = |x|$ where the function isn't differentiable?

-Dan
To shorten the below: no.

If something is not differentiable only when it at the very middle (where the right and left come together, usually in examples as (0,0,) of cusps, corners, vertical tangents, and a non-removable discontinuity, then I cannot think or see, using a graphing calculator, any points.

4. Originally Posted by Truthbetold
To shorten the below: no.

If something is not differentiable only when it at the very middle (where the right and left come together, usually in examples as (0,0,) of cusps, corners, vertical tangents, and a non-removable discontinuity, then I cannot think or see, using a graphing calculator, any points.
$f(x) = |x|$

Consider x near 0. When x approaches 0 from the left, the first derivative is -1. When x approaches 0 from the right, the first derivative is 1. So the derivative of |x| does not exist at x = 0.

We can make a similar argument for $f(x) = sin(|x|) - 1$. See the graph below. Notice the cusp at x = 0.

-Dan