6. The function is NOT differentiable at x=?
I'll do these for you:
We simply have to use the product rule, which states:
If and and , then:
So we just plug in:
Where we just used the chain rule for
Most teachers accept this answer, but if you need it any more simplified, here's how to go about it:
And that's that one.
6. We have the function:
We know that this is an absolute value function, and that at a single point it touches the x-axis and takes an immediate shot upwards to avoid negative values.
At this point is what is called a sharp point. Since it is a sharp point there, the derivative for the equation at that point doesn't exist.
That absolute value function can never be negative, so we can assume that the sharp point occurs at y = 0, so we just set the equation equal to 0:
Now we split the equation:
Now we solve each one:
They both seem to give us the same value, and that makes it clear that both sides of the absolute value graph converge to this sharp point, meaning that this function is not differentiable at