Linearization

The linearization or linear approximation of a differentiable function $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ at the point $\vec{a}=(a_1,\cdots ,a_n)is$

$$L(\vec{x})=f(\vec{a})+D_f(\vec{a})(\vec{x}-\vec{a})$$

where $D_f$ is the Total Derivative of $f$

in 1 dimension this simplifies to the more familiar linearization formula:

$$f(x)\approx f(a)+f^{\prime }(a)(x-a)=L(x)$$