Run the online derivative calculator:
webMathematica Explorations: Step-by-Step Derivatives
Derivative of: x*Tan[x]^(1/2)
With respect to: x
Gives:
This is valid for
For , you have
Derivative of: -x*Tan[x]^(1/2)
With respect to: x
which gives:
So we can combine these two into one expression for :
Well, if is a function of , in other words, , you can't do that directly. In this case,
So we compute the derivative as before to find . Only once you've done that can you put in to find what the value of is.
But because you have an absolute value, at the derivative doesn't exist. That's because the slope of the tangent line of changes sign at .
We just say it isn't differentiable at x=0 and exclude the point x=0 from the domain of the derivative. You don't need to bother doing anything else.
The domains of the function and its derivative are a bit complicated though (since only exists where ). Are you asked to state the domain of the derivative?
Actually, you are correct! That is the concept of a "weak derivative." This is a bit advanced though. You wouldn't have to see that stuff until you became a graduate student or a senior working in mathematics.
Don't let the following links blow your mind - just be aware that what you are talking about does exist.
Weak derivative - Wikipedia, the free encyclopedia
Subderivative - Wikipedia, the free encyclopedia