Approximation

1 min read Last updated Sat Jun 27 2026 08:46:52 GMT+0000 (Coordinated Universal Time)

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

f(x,y) \approx f(x_0,y_0) + f_x(x_0,y_0)\,\Delta x + f_y(x_0,y_0)\,\Delta y

Total Differential

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

dz = f_x(x,y)\,\text{d}x + f_y(x,y)\,\text{d}y

$dz$ 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 $f(x,y) = x^2 + xy$ .

At $(x_0, y_0) = (2, 4)$ :

$f(2,4) = 4 + 8 = 12$
$f_x = 2x + y \Rightarrow f_x(2,4) = 8$
$f_y = x \Rightarrow f_y(2,4) = 2$

\Delta f \approx 8(0.01) + 2(-0.02) = 0.08 - 0.04 = 0.04

f(2.01,\, 3.98) \approx 12.04

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