Definition 2.3.1. Suppose that $f$ is a differentiable function. We define the function $L$ such that
\[L(x)=f(a)+f'(a)(x-a),\]which is called the linearization of $f$ at $a$, or the linear approximation of $f$ at $a$ when estimating the approximated value of $f$ at $a$.
Definition 2.3.2. If $y=f(x)$, where $f$ is a differentiable function, the the differential $dx$ is an independent variable. The differential $dy$ is then defined in terms of $dx$ by the equation
\[dy=f'(x)\,dx.\]So $dy$ is a dependent variable; it depends on the values of $x$ and $dx$.