# 3. Differential calculus

### Definition 0

The number 

$$
f^{\prime}(a)=\lim \_{E \ni x \rightarrow a} \frac{f(x)-f(a)}{x-a}
$$

is called the derivative of the function $$f$$ at $$a$$.

### Definition 1 

A function $$f : E \to R$$ defined on a set $$E\subset R$$ is differentiable at a point x ∈ E that is a limit point of E if $$f (x + h) − f (x) = A(x)h + \alpha(x;h)$$, where $$h \mapsto A(x) h$$ is a linear function in $$h$$ and $$\alpha(x;h) = o(h)$$as $$h \to 0$$, $$x +h \in E$$.

### Definition 2

The function $$h \mapsto A(x) h$$ of *Definition 1*, which is linear in $$h$$, is called the differential of the function$$f : E \to \mathcal R$$ at the point $$x\in E$$ and is denoted $$\mathrm d f (x)$$ or $$\mathrm D f (x)$$. Thus, $$\mathrm d f (x)(h) = A(x)h$$.

We obtain

$$
\frac{\mathrm{d} f(x)(h)}{\mathrm{d} x(h)}=f^{\prime}(x)
$$

We denote the set of all such vectors by $$T\mathbb R(x\_0)$$ or $$T\mathbb R\_{x\_0}$$. Similarly, we denote by $$T\mathbb R(x\_0)$$ or $$T\mathbb R\_{y\_0}$$ the set of all displacement vectors from the point $$y\_0$$ along the y-axis. It can then be seen from the definition of the differential that the mapping

$$
\mathrm{d} f\left(x\_{0}\right): T \mathbb{R}\left(x\_{0}\right) \rightarrow T \mathbb{R}\left(f\left(x\_{0}\right)\right)
$$

### The derivative of an inverse function

If a function $$f$$ is differentiable at a point x0 and its differential $$\mathrm{d} f\left(x\_{0}\right): T \mathbb{R}\left(x\_{0}\right) \rightarrow T \mathbb{R}\left(y\_0\right)a$$ is invertible at that point, then the differential of the function $$f^{ −1}$$ inverse to $$f$$ exists at the point $$y\_0 = f (x\_0)$$ and is the mapping

$$
\mathrm{d} f^{-1}\left(y\_{0}\right)=\left\[\mathrm{d} f\left(x\_{0}\right)\right]^{-1}: T \mathbb{R}\left(y\_{0}\right) \rightarrow T \mathbb{R}\left(x\_{0}\right)
$$

inverse to  $$\mathrm{d} f\left(x\_{0}\right): T \mathbb{R}\left(x\_{0}\right) \rightarrow T \mathbb{R}\left(y\_0\right)a$$ .

### The derivative of some common function formula

1. $$(C)^{\prime}=0$$
2. $$\left(x^{\mu}\right)^{\prime}=\mu x^{\mu-1}$$
3. $$(\sin x)^{\prime}=\cos x$$
4. $$(\cos x)^{\prime}=-\sin x$$
5. $$(\tan x)^{\prime}=\sec ^{2} x$$
6. $$(\cot x)^{\prime}=-\csc ^{2} x$$
7. $$(\sec x)^{\prime}=\sec x \tan x$$
8. $$(\csc x)^{\prime}=-\csc x \cot x$$
9. $$\left(a^{x}\right)^{\prime}=a^{x} \ln a \quad(a>0, a \neq 1)$$
10. $$\left(\mathrm{e}^{x}\right)^{\prime}=\mathrm{e}^{x}$$
11. $$\left(\log \_{a} x\right)^{\prime}=\frac{1}{x \ln a}(a>0, a \neq 1)$$
12. $$(\ln x)^{\prime}=\frac{1}{x}$$
13. $$(\arcsin x)^{\prime}=\frac{1}{\sqrt{1-x^{2}}}$$
14. $$(\arccos x)^{\prime}=-\frac{1}{\sqrt{1-x^{2}}}$$
15. $$(\arctan x)^{\prime}=\frac{1}{1+x^{2}}$$
16. $$(\operatorname{arccot} x)^{\prime}=-\frac{1}{1+x^{2}}$$ 
17. $$(\sinh x)'= \cosh x$$ 
18. $$(\cosh x)'=\sinh x$$
19. $$(\tanh x)' =\frac{1}{\cosh ^{2} x}$$
20. $$(\operatorname{coth} x)' =-\frac{1}{\sinh ^{2} x}$$
21. $$(\operatorname{arsinh} x)'=\left(\ln \left(x+\sqrt{1+x^{2}}\right)\right)' = \frac{1}{\sqrt{1+x^{2}}}$$
22. $$(\operatorname{arcosh} x)'=(\ln \left(x \pm \sqrt{x^{2}-1}\right))'= \pm \frac{1}{\sqrt{x^{2}-1}}$$
23.  $$(\operatorname{artanh} x)'=(\frac{1}{2} \ln \frac{1+x}{1-x})' = \frac{1}{1-x^{2}}$$
24.  $$(\operatorname{arcoth} x)'=(\frac{1}{2} \ln \frac{x+1}{x-1})' = \frac{1}{x^{2}-1}$$

### theorem Leibniz

## L'Hôpital's rule

The theorem states that for functions $$f$$ and $$g$$ which are differentiable on an open interval $$I$$ except possibly at a point $$c$$ contained in $$I$$, if $${\displaystyle \lim {x\to c}f(x)=\lim {x\to c}g(x)=0{\text{ or }}\pm \infty ,}$$$${\displaystyle \lim {x\to c}f(x)=\lim {x\to c}g(x)=0}$$or $$\pm \infty$$ ,and $$g'(x)\neq 0$$$$g'(x)\neq 0$$ for all $$x$$ in $$I$$ with $$x \ne c$$, and $$\lim\_{x\to c}{\frac {f'(x)}{g'(x)}}$$ exists, then 

Suppose $$f,g$$are the function that defined in the set $$X$$, and also have the $$n\_{th}$$ derivative, s.t.

$$
(f g)^{(n)}(x)=\sum\_{k=0}^{n}\left(\begin{array}{l}
n \\
k
\end{array}\right) f^{(k)}(x) g^{(n-k)}(x)
$$

$$
\lim\_{x\to c}{\frac {f(x)}{g(x)}}=\lim\_{x\to c}{\frac {f'(x)}{g'(x)}}
$$

## mean value theorem

### Taylor's theorem

### Rolle's theorem

Let $$k \geq 1$$ be an integer and let the function $$f :\mathbb R\to\mathbb R$$ be $$k$$ times differentiable at the point $$a \in\mathbb R$$. Then there exists a function $$R\_k : \mathbb R \to\mathbb R$$ such that ,

If a real-valued function $$f$$ is continuous on a closed interval $$\[a, b]$$, differentiable on the open interval $$(a, b)$$, and $$f (a) = f (b)$$, then there exists at least one $$c$$ in the open interval $$(a, b)$$ such that $$f'(c)=0$$.

$$
f(a+x)=f(a)+f'(a)(x)+{\frac {f''(a)}{2!}}x^{2}+\cdots +{\frac {f^{(k)}(a)}{k!}}x^{k}+R\_k(x;a)
$$

### Lagrange’s Mean Value Theorem

and,

If a real-valued function $$f$$ is continuous on a closed interval $$\[a, b]$$, differentiable on the open interval $$(a, b)$$, then there is exists at least one $$c$$ in the open interval $$(a, b)$$ such that 

$$
f(b)−f(a)=f′(c)(b−a)
$$

### Cauchy’s Mean Value Theorem

If a real-valued function $$f$$ is continuous on a closed interval $$\[a, b]$$, differentiable on the open interval $$(a, b)$$, and $$g ' ( x ) \ne 0$$ for all $$x \in ( a , b )$$ , then there is exists at least one $$c$$ in the open interval $$(a, b)$$ such that 

$$
\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{\prime}(c)}{g^{\prime}(c)}
$$

$$
R\_{k}(x;a)=\int\_{a}^{a+x} \frac{f^{(k+1)}(t)}{k !}(a+x-t)^{k} \mathrm d t
$$

#### prove:

$$
\begin{aligned}
R\_{k}(x;a)=&\int\_{a}^{a+x} \frac{f^{(k+1)}(t)}{k !}(a+x-t)^{k} \mathrm d t\\
\=& \frac{-1}{k !}\int\_a^{a+x}f^{(k+1)}(t)(a+x-t)^k\mathrm d( a+x-t)\\
\=&\frac{-1}{(k+1) !}\int\_a^{a+x}f^{(k+1)}(t)\mathrm d( a+x-t)^{k+1}\\
\=&\frac{-1}{(k+1) !}f^{(k+1)}(t)( a+x-t)^{k+1}|*{a}^{a+x}+\frac{-1}{(k+1) !}\int\_a^{a+x}( a+x-t)^{k+1}\mathrm df^{(k+1)}(t)\\
\=&\frac{f^{(k+1)}(a)}{(k+1) !}x^{k+1}+R*{k+1}(x;a)
\end{aligned}
$$

q.e.d

#### remainder term

using little $$o$$ notation, $$R\_{k}(x;a)=o\left(|x|^{k}\right), \quad x \rightarrow 0$$(The Peano remainder term)

The Lagrange form remainder term( Mean-value forms)

$$
R\_{n}(x;a)=\frac{f^{(n+1)}(\theta)}{(n+1) !}x^{n+1}\quad (\theta\in(a,a+x))
$$

$$\blacksquare$$