(I'm doing to use d/dv etc for partial derivatives for ease of typing.)
dW/dv doesn't give you can approximation for W when you change v but it gives you the rate of how fast W changes.
If we pretend for a second that W were a function of only v, then we could approximate W(v-1) using the tangent line. (To do this, we use dW/dv(v) btw, not dW/dv(v-1).)
In this situation, W is a function of v and T, but since in this case we are holding T constant, the calculation is still the same.
I'm not sure what you're covering in your class right now, but this also ends up being the same calculation as if you used the tangent plane to make an approximation.