An indeterminate expression when calculating derivative.

Good afternoon,

f(x) = x * |cos(pi/x)| I want to know the value of df/dx (2/3) (right and left hand derivatives)

With a standard algorithm, I calculate it this way: d/dx x * |cos(pi/x)| = 1 * |cos(pi/x)| + x * (|cos(pi/x)|/(cos(pi/x))) * (-sin(pi/x)) * (-pi/x^2) = |cos(pi/x)| * (1 + tg(pi/x) * pi / x)

However, tg(3*pi/2) does not have value!

My questions are:

1. How is it possible that x * |cos(pi/x)| is differentiable in 2/3 but above expression is indeterminate?
2. I was able to solve this by definition (result is 3pi/2 from the right and -3pi/2 from the left). Is there another way of solving this?

Solution 1:

1. $f(x)$ is not differentiable at 2/3 because the left and right hand derivatives differ.

2. You can solve this by noticing that for $x$ slightly smaller than $2/3$, $\cos(\pi/x) > 0$, so you can take $f(x)=x\cos(\pi/x)$ and differentiate that. On the other hand, for $x$ slightly bigger than $2/3$, $\cos(\pi/x)<0$, so that you can differentiate $f(x)=-x\cos(\pi/x)$. Notice that this will be the negative of the previous value.