Approximation

If $f(x,y)$ is differentiable at $(x_0,y_0)$ then:

Total Differential

For $z = f(x,y)$, the total differential is:

$dz$ gives the exact change in the tangent plane for increments $dx$ and $dy$. It approximates $\Delta z = f(x+dx, y+dy) - f(x,y)$ when $dx$ and $dy$ are small.

Example

Approximate $f(2.01,\, 3.98)$ where $f(x,y) = x^2 + xy$.