total differential
When you nudge several inputs by tiny amounts at once, the total differential tells you how much the output moves to first order. It is the multivariable bookkeeping that adds up each variable's separate contribution: each input you change pushes the output by its partial derivative times its own little step, and you simply sum those pushes.
For f(x, y) the total differential is df = (partial f / partial x) dx + (partial f / partial y) dy, and for more variables you add one term per variable. Read it as a recipe: the change in f is approximately the sensitivity of f to x times the change in x, plus the sensitivity to y times the change in y. The symbols dx and dy are the (small) independent changes you impose; df is the resulting estimated change. This is the same content as the total derivative, written as a formula you can plug numbers into rather than as an abstract linear map. The approximation is exact in the limit of infinitesimal steps and accurate to first order for genuinely small ones; the error is second order in the step sizes.
The total differential is the computational core of error propagation, linear approximation, and related rates in several variables. An engineer measuring a quantity that depends on several uncertain inputs uses df to convert input tolerances into an output tolerance; a physicist uses it to write how a thermodynamic potential changes as several state variables shift. Whenever you see a statement like 'a small change in the radius and height changes the volume by about ...,' the total differential is doing the arithmetic underneath.
A cylinder's volume is V = pi r^2 h, so dV = 2 pi r h dr + pi r^2 dh. With r = 5, h = 10, increasing r by dr = 0.1 and h by dh = 0.2 changes V by about 2 pi (5)(10)(0.1) + pi (25)(0.2) = 10 pi + 5 pi = 15 pi.
Each variable contributes its partial derivative times its own small change; the contributions simply add.
The total differential is a first-order estimate, not an exact change; it ignores second-order terms. For large input changes it can be far off, and it is only valid when the function is actually differentiable at the point.