1. Definition
If f is differentiable, df = f_x dx + f_y dy. For small increments, Δf≈df: the tangent plane locally replaces the surface.
2. Symbols
| f_x,f_y | partial derivatives |
| dx,dy | increments |
| df | total differential |
| Δf | actual change |
3. How it works
- Numerically find f_x, f_y;
- Compute df;
- Also compute Δf to compare;
- Smaller increments give better agreement.
4. Steps
- Enter f(x,y);
- Enter point and increments;
- Click Calculate.
5. Example
Example: f=x²y at (1,1), dx=dy=0.1.
Solution: f_x=2, f_y=1, df=0.3. Actual Δf=1.331−1=0.331; df is the linear part.
6. Pitfalls
Smaller increments → df≈Δf;
df keeps only first-order terms;
Used in approximation and error propagation.