Method 3, in plain English

Speed is distance divided by time. You measured both. Neither measurement is perfect. Distance might be off by H8. Time might be off by H9. You want one number that says how wobbly the speed is. That number is δv.

Freeze one measurement. Slope the other. Combine the two wiggles like a hypotenuse. Drag the graphs. The Excel cells sit at the bottom.

Two knobs

If the measured distance is a little too big, the speed you calculate comes out a little too big. Drag distance on the graph. The point climbs the line.

If the measured time is a little too big, the speed you calculate comes out a little too small. Same trip, longer clock, smaller speed. Drag time. The line gets flatter and the point drops.

Both knobs can be wrong at once. Method 3 figures out how much each knob moves the speed, then combines those two moves.

The only new idea

You already know a derivative. It is a slope. "If this one input wiggles, how much does the output wiggle?" The graph above is that question. The dashed line is the slope at the dot. Drag x and watch the dashed line tilt.

Here there are two inputs. So you ask that slope question twice. First time, pretend time is stuck. Only distance can wiggle. Second time, pretend distance is stuck. Only time can wiggle.

That "pretend the other one is stuck" move is what people call a partial derivative. Same derivative you already know. One extra freeze.

Step 1. Freeze time

Time is now a constant. Speed is just (1 / time) times distance. A straight line through the origin. The teal triangle is rise over run. Rise / run is 1 / time.

Step 2. Freeze distance

Distance is now a constant. Speed is distance times time⁻¹. That is a curve, not a line. The dashed line is the slope at your measured time. It points down. Longer clock, smaller speed.

Power rule on time⁻¹ gives −time⁻². Multiply by the stuck distance. Slope is minus distance over time squared.

Step 3. Size of each wiggle

Slope times how wrong that measurement is.

Distance's effect on speed = (step 1 cell) × H8

Time's effect on speed = (step 2 cell) × H9

Call the step 1 cell A and the step 2 cell B if you want names. Then the two effects are A*H8 and B*H9. The graph shows those two Δv bars. Drag the uncertainties and the bars grow.

Step 4. Combine them

Do not add the two effects. Adding is method 1's "both mistakes go the worst way."

Method 3 treats them as independent, like the two legs of a right triangle. The uncertainty is the hypotenuse. Drag the sliders. The dashed line is the add-them-up answer. It is always longer.

Square, add, square root. Squaring deletes the minus, so the uncertainty comes out positive.

Step 5. Write the answer

Speed is the same formula you already used.

Put that in one cell. Put the SQRT in the cell next to it. That pair is v ± δv.

What to type, in order

1. Under the "method 3" label, pick two empty cells for the slopes. Type =1/G9 in the first. Type =-G8/(G9^2) in the second.
2. Check the numbers. First cell a bit over 0.6. Second cell around −3.7. If those are wrong, the rest will be wrong.
3. In a third cell type =G8/G9. That is the speed.
4. Next to it, type the SQRT formula using those two slope cells times H8 and H9.