total differential

The differential-notation version of linear approximation: $dz=f_x\,dx+f_y\,dy$ .

The total differential $dz = f_x\,dx + f_y\,dy$ is the same statement as linear approximation without committing to specific new coordinates $(x,y)$ : $dx$ and $dy$ are small, independent changes away from $(x_0,y_0)$ , and $dz$ is the tangent plane's resulting change in height — an estimate of the true $\Delta z = f(x_0+dx,y_0+dy)-f(x_0,y_0)$ that gets more accurate the smaller $dx,dy$ are, requiring no evaluation of $f$ at the new point at all.

Examples

If $x$ and $y$ are physical measurements with known uncertainty $dx$ , $dy$ , then $dz=f_x\,dx+f_y\,dy$ estimates how much that error propagates into a computed quantity.