Chapter 2: Derivatives and Differentials

2.1 derivative

derivative definition

Definition:
If $\lim_{\Delta x\to0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}$ exists, then $f$ is differentiable at $x_0$.
Alternative form:
$f^{\prime}(x_0)=\lim_{\Delta x\to0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}=\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$
Form 1: $f^{\prime}(x_0)=\lim_{\Delta x\to0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}$
Form 2: $f^{\prime}(x_0)=\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$

left and right derivative

The relationship between the left and right derivatives: differentiable <-->The left and right derivatives exist and are equal;

algebraic manipulation law

Assume that $u(x), \nu(x)$ can all be imported, then

$(u\pm v)^{\prime}=u^{\prime}\pm v^{\prime}$
$(uv)^{\prime}=u^{\prime}v+uv^{\prime}$
$(\frac\mu\nu)^{\prime}=\frac{\mu^{\prime}\nu-\nu^{\prime}\mu}{\nu^2}\quad(\nu\neq0)$

inverse function differentiate

Formula:
$$(f^{-1})^{\prime}(x)=\frac1{f^{\prime}(y)}\quad\frac{dy}{dx}=\frac1{dx}$$
Meaning:
function and inverse function are two functions, but they are the same curve;
Thereforewhen the function is continuity and monotonic, it can be known that its inverse function is also differentiable;

Chain Rule

Suppose $u=g(x)$ is differentiable at $x$ and $y=f(u)$ is differentiable at the corresponding $u$, then $y=f[g(x)]$ is differentiable at x.
then: $\frac{dy}{dx}=f^{\prime}(u)g^{\prime}(x)$
Other forms:
$$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$$

basic elementary functions

Basic
$\begin{aligned}&\quad(C)^{\prime}=0\quad&\quad(x^\alpha)^{\prime}=\alpha x^{\alpha-1}\\&\quad(a^x)^{\prime}=a^x\ln a\quad&\quad(e^x)^{\prime}=e^x\\&\quad(\log_ax)^{\prime}=\frac1{x\ln a}\quad&\quad(\ln|x|)^{\prime}=\frac1x\end{aligned}$
trigonometric functions
$(\sin x)'=\cos x\qquad(\cos x)'=-\sin x$
$\begin{aligned}(\tan x)^{\prime}&=\sec^2x&(\cot x)^{\prime}&=-\csc^2x\\\\(\sec x)^{\prime}&=\sec x\tan x&(\csc x)^{\prime}&=-\csc x\cot x\end{aligned}$
inverse function derivative
$\begin{aligned}&(\arcsin x)'=\frac1{\sqrt{1-x^2}}&&(\arccos x)'=-\frac1{\sqrt{1-x^2}}\\&(\arctan x)'=\frac1{1+x^2}&&(\operatorname{arccot}x)'=-\frac1{1+x^2}\end{aligned}$

Example: Let $y=(1+\sin x)^x$. Find $\mathrm{d}y\big|_{x=\pi}$.

Analysis
This is an exponent function type, logarithmic differentiation can be used;
Analysis
Logarithm first: $\ln y=x\ln(1+\sin x)$
Differentiate both sides: $\frac{y'}{y}=\ln(1+\sin x)+\frac{x\cos x}{1+\sin x}$.
Propose $y^{\prime}$, get the derivative function
bring in x=Π;

IMPORTANT CONCLUSION

Applicable occasions

Commonly find n-order derivative

Differential Definition

If $f(x_0+\Delta x)-f(x_0)=A\Delta x+o(\Delta x)$, then $f(x)$ is said to be differentiable at $x _0$ point $; $
$A\Delta x$ is called the derivative of $f(x)$ at the $x_0$ point
The first part is the main part of the function change
The second part is higher-order infinitesimal, which is the secondary part

The relationship between differentiability, differentiability, and continuity

Formula: Taylor's formula with Peano remainder

Suppose $f(x)$ is differentiable in $n$ order at $x_0$, then:
$$f(x)=\sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k+o((x-x_0)^n)$$
$$f\left(x\right)=\frac{f\left(x_{0}\right)}{0!}+\frac{f^{\prime}\left(x_{0}\right)}{1!}(x-x_{0})+\frac{f^{\prime\prime}\left(x_{0}\right)}{2!}(x-x_{0})^{2}+\ldots+\frac{f^{\left(n\right)}\left(x_{0}\right)}{n!}(x-x_{0})^{n}+o((x-x_0)^n)$$
The remainder involving $f^{(n+1)}(\xi)$ is the Lagrange form of the remainder; a remainder written as $o\!\left((x-x_0)^n\right)$ is the Peano form.

Formula: Taylor's formula with Peano remainder - Taylor expansion of zero points

$$\begin{gathered}\text{If }x_0=0,\text{ then}\\f(x)=f(0)+f^{\prime}(0)x+\frac{f^{\prime\prime}(0)}{2!}x^2+\cdots+\frac{f^{(n)}(0)}{n!}x^n+\frac{f^{(n+1)}(\theta x)}{(n+1)!}x^{n+1},\quad0<\theta<1.\\\text{This is the Maclaurin formula for }f(x).\end{gathered}$$

Formula: Taylor's formula with Lagrangian remainder

where $\xi$ is between $x$ and $x_0$, so
$$f(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime\prime}(a)}{2!}(x-a)^2+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^n+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_{0})^{n+1}$$