Find all values of x for which the function is differentiable (involves absolute value)

Question

Find all values of x for which the function is differentiable.
given f(x) = sin(|x|)-1, find all values of x for which the function is differentiable.

How do I solve the question above? My teacher explained it today, but I was utterly and completely lost. If you could provide a relatively understandable explanation and a step-by-step solution, it would be much appreciated.
Thank you in advance!!!

Answer 1

Substitute |X| with sqrt(X^2)   <-- the positive square root

 

f(x) = sin( sqrt((x)^2)) - 1

 

I used the extra parenthesis to handle negative numbers.

For example

  f(-3) = sin (sqrt ( (-3)^2 ) - 1

 

          = sin (sqrt(9) ) - 1

           = sin (3) -1

 

 

Now for the derivative:

 

 y = f(x) = sin( sqrt((x)^2))-1

 

             

By chain rule, recursively, we need derivative of sqrt( (x)^2 )

 which means we will need the derivative of x^2

 

y' = cos( sqrt((x)^2))  *    ((x)^2)^(-1/2)  *   (2x)

 

   = cos ( sqrt((x)^2) * (2x) / ((x)^2)^(1/2)

 

   = cost(sqrt((x)^2)*(2x) /  sqrt((x)^2))

 

 

So the derivative is undefined at x=0