Yes. You get this answer via implicit differentiation. A fun way to check:
As the slope is -2/3, (1.015, 1.01) is on the tangent line.
Calculate f(1.015, 1.0099) + 1/6 and
f(1.015, 1.0101) + 1/6
If your slope is correct, you'll see that one of the values is greater than one and the other is less, verifying the slope. That happens here.
Another option would be to vary x by a little from 1 and use Excel's Goal Seek on y to get the function to equal -1/6, then calculate the slope from (1, 1) to that point.