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 $f^{\prime }(a)=A$.

Differential in two variables

(from: Calculus: Early transcendentals (p. 455), by J. Stewart, D. K. Clegg, and S. Watson, 2020, Cengage Learning)

Let $f(x,y)$ be a differentiable function at a point $(a,b)$:

• The total differential (or differential) of $f$ at $(a,b)$ is defined as: \begin{align*} dz=df&=f_x(a,b)\,dx+f_y(a,b)\,dy \\ &= \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy \end{align*}
• $dz=\nabla z\cdot d\mathbf{r}$ where:
  • $\nabla z$ is the gradient of $z$
  • $d\mathbf{r}=(dx,dy)$