Differential

• Let $f(x)$ be a differentiable function at a point $a$:
  • The linear function $d{f}_a(dx)=f^{\prime }(a)dx$ is called the differential of $f$ at $a$.
    • Denoted by $dy$ and $df$ (where $a$ is known)
      • (e.g. if $f(x)=\sin (x)$, then $dy=\cos (a)dx$)
    • $dx$ is the independent variable of the function $d{f}_a=dy$
      • It is called the differential of $x$ (or increment of $x$)
      • It is also denoted by $h$ or $\Delta x$.
    • $dy$ is dependent variable (on both $dx$ and $a$)
      • It is called the differential of $y$
    • $\Delta y=f(a+dx)-f(a)$
    • $dy=f^{\prime }(a)dx\approx \Delta y$ when $dx$ is small
    • $\epsilon =\Delta y-dy=f(a+dx)-f(a)-f^{\prime }(a)dx$ is called the error of the approximation.
    • $\underset{dx\to 0}{\lim }\frac{\Delta y-dy}{dx}=\underset{dx\to 0}{\lim }\frac{f(a+dx)-f(a)-f^{\prime }(a)dx}{dx}=0$
  • The function $r(x)=f(x)-L(x)$ is called the error of the approximation of $f$ at $a$.
  • The function $L(x)=f(a)+f^{\prime }(a)(x-a)$ is called the linearization of $f$ at $a$
  • The approximation of $f$ by $L$, that is $f(x)\approx L(x)$, is the (standard) linear approximation (or tangent line approximation) of $f$ at $a$
  • The point $x=a$ is the center of the approximation
  • $\underset{x\to a}{\lim }\frac{r(x)}{x-a}=0$
• Let $f(x)$ be a function defined in a neighborhood of $a$.
  • If $f$ is differentiable at $a$, then $f(x)=f(a)+f^{\prime }(a)(x-a)+r(x)$, where $\underset{x\to a}{\lim }\frac{r(x)}{x-a}=0$
  • If there exists $A\in \mathbb{R}$ and function $r(x)$ such that $f(x)=f(a)+A\cdot (x-a)+r(x)$ and $\underset{x\to a}{\lim }\frac{r(x)}{x-a}=0$, then $f$ is differentiable at $a$ and $f^{\prime }(a)=A$.