Kinetic energy with variable forces

Now what happens to the eq. 1.4 when we have variable forces? Basically nothing. It can be see as follows. Still using this staircase figure, we see that along a step we can apply eq. 1.4. The incremental work moving a distance $\Delta x$ is still the change in kinetic energy.

Now if we move two steps the total work is just the sum of the work from the individual steps $W_1 + W_2$, say. This is just the sum of the two changes in kinetic energy. But the total change in kinetic energy is just the sum of the two changes. So that means that for two steps eq. 1.4 is still true. So generalizing this, you can see that it will be true for the addition any number of steps. So eq. 1.4 is true for any variable force in one dimension.