Change one input while holding another fixed
How does a two-input function have more than one local rate?
Entering midway? A derivative measures local change in one variable. Holding another variable fixed turns a two-input function into a one-input slice.
Start with a concrete question
Inspect slices of F(x,y)=x²+y²+cxy, with a coupling parameter c. The plot shows two slices, not a perspective surface.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Will the x and y partial derivatives be equal at every point?
Change one thing. Inspect what happens.
- Keep y fixed and vary x; identify which terms change.
- Compare the x and y slices at an asymmetric point such as (1,−2).
- Set coupling zero, then restore it and explain why one coordinate affects the other partial.
- Predict a small two-coordinate change using both partials and state why it is only a local estimate.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Two one-variable slices of the same surface
Horizontal: Varied coordinate t (other held fixed). Vertical: Surface value.
- 1. Vary x, y=1 (solid line)
- 2. Vary y, x=1 (dashed line)
- 3. Selected slice points (points)
Axes use the displayed ranges.
Read plotted values as a table
| Series | Varied coordinate t (other held fixed) | Surface value |
|---|---|---|
| Vary x, y=1 | -2 | 3 |
| Vary x, y=1 | -1.95 | 2.853 |
| Vary x, y=1 | -1.9 | 2.71 |
| Vary x, y=1 | -1.85 | 2.573 |
| Vary x, y=1 | -1.8 | 2.44 |
| Vary x, y=1 | -1.75 | 2.313 |
| Vary x, y=1 | -1.7 | 2.19 |
| Vary x, y=1 | -1.65 | 2.072 |
| Vary x, y=1 | -1.6 | 1.96 |
| Vary x, y=1 | -1.55 | 1.853 |
| Vary x, y=1 | -1.5 | 1.75 |
| Vary x, y=1 | -1.45 | 1.653 |
| Vary x, y=1 | -1.4 | 1.56 |
| Vary x, y=1 | -1.35 | 1.473 |
| Vary x, y=1 | -1.3 | 1.39 |
| Vary x, y=1 | -1.25 | 1.313 |
| Vary x, y=1 | -1.2 | 1.24 |
| Vary x, y=1 | -1.15 | 1.172 |
| Vary x, y=1 | -1.1 | 1.11 |
| Vary x, y=1 | -1.05 | 1.052 |
| Vary x, y=1 | -1 | 1 |
| Vary x, y=1 | -0.95 | 0.9525 |
| Vary x, y=1 | -0.9 | 0.91 |
| Vary x, y=1 | -0.85 | 0.8725 |
| Vary x, y=1 | -0.8 | 0.84 |
| Vary x, y=1 | -0.75 | 0.8125 |
| Vary x, y=1 | -0.7 | 0.79 |
| Vary x, y=1 | -0.65 | 0.7725 |
| Vary x, y=1 | -0.6 | 0.76 |
| Vary x, y=1 | -0.55 | 0.7525 |
| Vary x, y=1 | -0.5 | 0.75 |
| Vary x, y=1 | -0.45 | 0.7525 |
| Vary x, y=1 | -0.4 | 0.76 |
| Vary x, y=1 | -0.35 | 0.7725 |
| Vary x, y=1 | -0.3 | 0.79 |
| Vary x, y=1 | -0.25 | 0.8125 |
| Vary x, y=1 | -0.2 | 0.84 |
| Vary x, y=1 | -0.15 | 0.8725 |
| Vary x, y=1 | -0.1 | 0.91 |
| Vary x, y=1 | -0.05 | 0.9525 |
| Vary x, y=1 | 0 | 1 |
| Vary x, y=1 | 0.05 | 1.052 |
| Vary x, y=1 | 0.1 | 1.11 |
| Vary x, y=1 | 0.15 | 1.172 |
| Vary x, y=1 | 0.2 | 1.24 |
| Vary x, y=1 | 0.25 | 1.313 |
| Vary x, y=1 | 0.3 | 1.39 |
| Vary x, y=1 | 0.35 | 1.473 |
| Vary x, y=1 | 0.4 | 1.56 |
| Vary x, y=1 | 0.45 | 1.653 |
| Vary x, y=1 | 0.5 | 1.75 |
| Vary x, y=1 | 0.55 | 1.852 |
| Vary x, y=1 | 0.6 | 1.96 |
| Vary x, y=1 | 0.65 | 2.072 |
| Vary x, y=1 | 0.7 | 2.19 |
| Vary x, y=1 | 0.75 | 2.313 |
| Vary x, y=1 | 0.8 | 2.44 |
| Vary x, y=1 | 0.85 | 2.573 |
| Vary x, y=1 | 0.9 | 2.71 |
| Vary x, y=1 | 0.95 | 2.853 |
| Vary x, y=1 | 1 | 3 |
| Vary x, y=1 | 1.05 | 3.152 |
| Vary x, y=1 | 1.1 | 3.31 |
| Vary x, y=1 | 1.15 | 3.472 |
| Vary x, y=1 | 1.2 | 3.64 |
| Vary x, y=1 | 1.25 | 3.813 |
| Vary x, y=1 | 1.3 | 3.99 |
| Vary x, y=1 | 1.35 | 4.173 |
| Vary x, y=1 | 1.4 | 4.36 |
| Vary x, y=1 | 1.45 | 4.553 |
| Vary x, y=1 | 1.5 | 4.75 |
| Vary x, y=1 | 1.55 | 4.952 |
| Vary x, y=1 | 1.6 | 5.16 |
| Vary x, y=1 | 1.65 | 5.372 |
| Vary x, y=1 | 1.7 | 5.59 |
| Vary x, y=1 | 1.75 | 5.813 |
| Vary x, y=1 | 1.8 | 6.04 |
| Vary x, y=1 | 1.85 | 6.273 |
| Vary x, y=1 | 1.9 | 6.51 |
| Vary x, y=1 | 1.95 | 6.753 |
| Vary x, y=1 | 2 | 7 |
| Vary y, x=1 | -2 | 3 |
| Vary y, x=1 | -1.95 | 2.853 |
| Vary y, x=1 | -1.9 | 2.71 |
| Vary y, x=1 | -1.85 | 2.573 |
| Vary y, x=1 | -1.8 | 2.44 |
| Vary y, x=1 | -1.75 | 2.313 |
| Vary y, x=1 | -1.7 | 2.19 |
| Vary y, x=1 | -1.65 | 2.072 |
| Vary y, x=1 | -1.6 | 1.96 |
| Vary y, x=1 | -1.55 | 1.853 |
| Vary y, x=1 | -1.5 | 1.75 |
| Vary y, x=1 | -1.45 | 1.653 |
| Vary y, x=1 | -1.4 | 1.56 |
| Vary y, x=1 | -1.35 | 1.473 |
| Vary y, x=1 | -1.3 | 1.39 |
| Vary y, x=1 | -1.25 | 1.313 |
| Vary y, x=1 | -1.2 | 1.24 |
| Vary y, x=1 | -1.15 | 1.172 |
| Vary y, x=1 | -1.1 | 1.11 |
| Vary y, x=1 | -1.05 | 1.052 |
| Vary y, x=1 | -1 | 1 |
| Vary y, x=1 | -0.95 | 0.9525 |
| Vary y, x=1 | -0.9 | 0.91 |
| Vary y, x=1 | -0.85 | 0.8725 |
| Vary y, x=1 | -0.8 | 0.84 |
| Vary y, x=1 | -0.75 | 0.8125 |
| Vary y, x=1 | -0.7 | 0.79 |
| Vary y, x=1 | -0.65 | 0.7725 |
| Vary y, x=1 | -0.6 | 0.76 |
| Vary y, x=1 | -0.55 | 0.7525 |
| Vary y, x=1 | -0.5 | 0.75 |
| Vary y, x=1 | -0.45 | 0.7525 |
| Vary y, x=1 | -0.4 | 0.76 |
| Vary y, x=1 | -0.35 | 0.7725 |
| Vary y, x=1 | -0.3 | 0.79 |
| Vary y, x=1 | -0.25 | 0.8125 |
| Vary y, x=1 | -0.2 | 0.84 |
| Vary y, x=1 | -0.15 | 0.8725 |
| Vary y, x=1 | -0.1 | 0.91 |
| Vary y, x=1 | -0.05 | 0.9525 |
| Vary y, x=1 | 0 | 1 |
| Vary y, x=1 | 0.05 | 1.052 |
| Vary y, x=1 | 0.1 | 1.11 |
| Vary y, x=1 | 0.15 | 1.172 |
| Vary y, x=1 | 0.2 | 1.24 |
| Vary y, x=1 | 0.25 | 1.313 |
| Vary y, x=1 | 0.3 | 1.39 |
| Vary y, x=1 | 0.35 | 1.473 |
| Vary y, x=1 | 0.4 | 1.56 |
| Vary y, x=1 | 0.45 | 1.653 |
| Vary y, x=1 | 0.5 | 1.75 |
| Vary y, x=1 | 0.55 | 1.852 |
| Vary y, x=1 | 0.6 | 1.96 |
| Vary y, x=1 | 0.65 | 2.072 |
| Vary y, x=1 | 0.7 | 2.19 |
| Vary y, x=1 | 0.75 | 2.313 |
| Vary y, x=1 | 0.8 | 2.44 |
| Vary y, x=1 | 0.85 | 2.573 |
| Vary y, x=1 | 0.9 | 2.71 |
| Vary y, x=1 | 0.95 | 2.853 |
| Vary y, x=1 | 1 | 3 |
| Vary y, x=1 | 1.05 | 3.152 |
| Vary y, x=1 | 1.1 | 3.31 |
| Vary y, x=1 | 1.15 | 3.472 |
| Vary y, x=1 | 1.2 | 3.64 |
| Vary y, x=1 | 1.25 | 3.813 |
| Vary y, x=1 | 1.3 | 3.99 |
| Vary y, x=1 | 1.35 | 4.173 |
| Vary y, x=1 | 1.4 | 4.36 |
| Vary y, x=1 | 1.45 | 4.553 |
| Vary y, x=1 | 1.5 | 4.75 |
| Vary y, x=1 | 1.55 | 4.952 |
| Vary y, x=1 | 1.6 | 5.16 |
| Vary y, x=1 | 1.65 | 5.372 |
| Vary y, x=1 | 1.7 | 5.59 |
| Vary y, x=1 | 1.75 | 5.813 |
| Vary y, x=1 | 1.8 | 6.04 |
| Vary y, x=1 | 1.85 | 6.273 |
| Vary y, x=1 | 1.9 | 6.51 |
| Vary y, x=1 | 1.95 | 6.753 |
| Vary y, x=1 | 2 | 7 |
| Selected slice points | 1 | 3 |
| Selected slice points | 1 | 3 |
Follow the calculation
- F(x,y)=x²+y²+1xy; at (1,1), F=3.
- Holding y=1 fixed, the x slice is t²+1+(1)t. Its slope at t=1 is 2×1+1×1=3.
- Holding x=1 fixed, the y slice is 1+t²+(1)t. Its slope at t=1 is 2×1+1×1=3.
- The gradient collects both partials: (3, 3). For a small proposed change (Δx,Δy), ΔF≈3Δx+3Δy.
A partial derivative changes one input while holding the others fixed. Different slices through the same point can have different slopes.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Start with the meaning
A derivative measures local change in one variable. Holding another variable fixed turns a two-input function into a one-input slice.
Inspect slices of F(x,y)=x²+y²+cxy, with a coupling parameter c. The plot shows two slices, not a perspective surface.
A worked explanation
The partial derivative in x treats y as a constant. For this function, ∂F/∂x=2x+cy and ∂F/∂y=2y+cx.
At (1,1) and c=1, F=3 and both partials equal 3. At (1,−2), the x partial is zero and y partial is −3, despite belonging to the same surface.
The gradient is the vector of partial derivatives. Dotting it with a small change vector predicts first-order output change. Higher-order terms remain for finite changes.
Symbols and relationship
∇F=(2x+cy,2y+cx); ΔF≈(∂F/∂x)Δx+(∂F/∂y)Δy. ∂ means vary one input while holding others fixed.
Use the graph and trace together
The graph and calculations use your current inputs. Read axis labels and units before comparing positions or slopes. Change one input, predict the effect, and use the numerical trace to verify the plotted behavior. A displayed curve is an illustration; the formula and its assumptions determine which conclusions are justified.
Scope and edge cases
- A smooth quadratic surface with bounded coupling, not an empirical cost model.
- The two plotted horizontal axes both use t but hold different other coordinates fixed; they are slices, not two optimization paths.
Try a fresh case
For G(x,y)=x²+3xy at (2,4), what is ∂G/∂x?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Evaluate nearby points changing only one coordinate at a time.
College investigation
Derive the Hessian and compare the exact quadratic remainder with the first-order approximation.
Workplace investigation
Document which inputs are held fixed in a two-factor sensitivity analysis.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.