💡 Let $D\subseteq \mathbb{R}$, $f:D⟶\mathbb{R}$ and let $x_0\in D$ be an accumulation point of $D$.

$f$ is differentiable in $x_0$ if the limit

$$\underset{x\to x_0}{\lim }\frac{f(x)-f(x_0)}{x-x_0}$$

exists. If this is the case, then this limit is denoted by by $f^{\prime }(x_0)$.

Oftentimes it is useful to replace $x$ by $x=x_0+h$ in this definition, such that

$$\underset{h\to 0}{\lim }\frac{f(x_0+h)-f(x_0)}{h}.$$

💡 $\frac{f(x)-f(x_0)}{x-x_0}$ is the the slope of the line through $(x_0,f(x_0)),(x,f(x))$. If $f^{\prime }(x_0)$ exists intuitively it corresponds to the fact that the family of lines through $(x_0,f(x_0)),(x,f(x))$ for $x\ne x_0$, $x\to x_0$ as a “limit” approaches the tangent line to the graph of $f$ in $(x_0,f(x_0))$.

📖 (Weierstrass, 1861).

Let $f:D⟶\mathbb{R}$ and let $x_0\in D$ be an accumulation point of $D$. The following statements are equivalent:

1. $f$ is differentiable in $x_0$,
2. There exist $c\in \mathbb{R}$ and $r:D⟶\mathbb{R}$ with:
   1. $f(x)=f(x_0)+c(x-x_0)+r(x)(x-x_0)$
   2. $r(x_0)=0$ and r is continuous in $x_0$.

If this applies, then $c=f^{\prime }(x_0)$ is uniquely determined.

💡 Formulating the differentiability of $f$ using

$$f(x)=f(x_0)+f^{\prime }(x_0)(x-x_0)+r(x)(x-x_0)$$

and the continuity of $r$ in $x_0$ has the advantage that it does not contain a limit. In particular

$$y=f(x_0)+f^{\prime }(x_0)(x-x_0)$$

is the equation of the tangent line of $f$ in the point $(x_0,f(x_0))$. Furthermore, this can be simplified to

$$\varphi (x)=f^{\prime }(x_0)+r(x).$$

📖 A function $f:D⟶\mathbb{R}$ is differentiable in $x_0$ if and only if there exists a function $\varphi :D⟶\mathbb{R}$ that is continuous in $x_0$ and such that

$$f(x)=f(x_0)+\varphi (x)(x-x_0)\forall x\in D.$$

In this case it holds that $\varphi (x_0)=f^{\prime }(x_o).$

📎 Let $f:D⟶\mathbb{R}$ and let $x_0\in D$ be an accumulation point of $D$. If $f$ is differentiable in $x_0$ then $f$ is continuous in $x_0$.

💡 $f:D⟶\mathbb{R}$ is differentiable in $D$, if for every accumulation point $x_0\in D$, $f$ is differentiable in $x_0$,

💡 Examples:

1. $\exp :\mathbb{R}⟶\mathbb{R}$ is differentiable in $\mathbb{R}$ and ${\exp }^{\prime }=\exp$.
2. ${\sin }^{\prime }=\cos$ and ${\cos }^{\prime }=-\sin$.

📖 Let $D\subseteq \mathbb{R}$, $x_0\in D$ be an accumulation point of $D$ and let $f,g:D⟶\mathbb{R}$ be differentiable in $x_0$. Then the following hold:

1. $f+g$ is differentiable in $x_0$ and

   $$(f+g{)}^{\prime }(x_0)=f^{\prime }(x_0)+g^{\prime }(x_0)$$

2. $f\cdot g$ is differentiable in $x_0$ and

   $$(f\cdot g{)}^{\prime }(x_0)=f^{\prime }(x_0)g(x_0)+f(x_0)g^{\prime }(x_0).$$

3. If $f(x_0)\ne 0$ then $\frac{f}{g}$ is differentiable in $x_0$ and

   $$(\frac{f}{g}{)}^{\prime }(x_0)=\frac{f^{\prime }(x_0)g(x_0)-f(x_0)g^{\prime }(x_0)}{g(x_0{)}^2}.$$

💡 Examples:

1. $(x^n{)}^{\prime }=n{x}^{n-1}\forall x\in \mathbb{R},n\ge 1$

2. The tangent function

   $$\tan x=\frac{\sin x}{\cos x},x\notin \frac{\pi }{2}+\pi \mathbb{Z}$$

   is differentiable on its domain and

   $${\tan }^{\prime }(x)=\frac{1}{{\cos }^{2}x}.$$

3. The cotangent function

   $$\cot x=\frac{\cos x}{\sin x},x\notin \pi \mathbb{Z}$$

   is differentiable on its domain and

   $${\cot }^{\prime }(x)=-\frac{1}{{\sin }^2(x)}.$$

📖 Let $D,E\subseteq \mathbb{R}$ and let $x_0\in D$ be an accumulation point. Let $f:D⟶E$ be a function that is differentiable in $x_0$ such that $y_0=f(x_0)$ is an accumulation point of $E$, and let $g:E⟶\mathbb{R}$ be a function that is differentiable in $y_0$. Then $g\circ f:D⟶\mathbb{R}$ is differentiable in $x_0$ and

$$(g\circ f{)}^{\prime }(x_0)=g^{\prime }(f(x_0))f^{\prime }(x_0).$$

📎 Let $f:D⟶E$ be a bijective function and let $x_0\in D$ be an accumulation point. Let us assume that $f$ is differentiable in $x_0$ and $f^{\prime }(x_0)\ne 0$. Assume also that $f^{-1}$ is continuous in $y_0=f(x_0)$. Then $y_0$ is an accumulation point of $E$, $f^{-1}$ is differentiable in $y_0$ and

$$(f^{-1}{)}^{\prime }(y_0)=\frac{1}{f^{\prime }(x_0)}.$$

💡 Examples:

1. The derivative of $\ln :(0,+\infty )⟶\mathbb{R}$ is

   $${\ln }^{\prime }(x)=\frac{1}{x}.$$

   For all $x\in \mathbb{R}$ it holds that:

   $$\ln (\exp (x))=x.$$

   Applying the chain rule to $f(x)=\exp x$ and $g(y)=\ln y$ we get by taking the derivative:

   $${\ln }^{\prime }(\exp x){\exp }^{\prime }(x)=1\forall x\in \mathbb{R}.$$

   Since ${\exp }^{\prime }(x)=\exp (x)\forall x\in \mathbb{R}$ it follows that

   $${\ln }^{\prime }(\exp x)\exp x=1\forall x\in \mathbb{R}$$

   and since $\exp :\mathbb{R}⟶(0,\infty )$ is bijective it follows that:

   $${\ln }^{\prime }(y)\cdot y=1\forall y\in (0,\infty ).$$

2. Let $a\in \mathbb{R}$; the derivative of the function

   $$\begin{array}{rl}(0,\infty )& ⟶\mathbb{R}\\ x& ⟼x^a\end{array}$$

   is $a{x}^{a-1}$.

   By definition: $x^a=\exp (a\ln x),x>0$. We apply the chain rule to $f(x)=a\ln x$ and $g(x)=\exp y$ and we get with $g^{\prime }(y)=\exp y$ and $f^{\prime }(x)=a\cdot \frac{1}{x}$,

   $$\begin{array}{rl}(x^a{)}^{\prime }& ={\exp }^{\prime }(a\ln x)\frac{a}{x}=\exp (a\ln x)\frac{a}{x}\\ & =x^a\frac{a}{x}=a{x}^{a-1}.\end{array}$$

3. $f:\mathbb{R}⟶\mathbb{R},x⟼x^3$ is bijectively differentiable in $\mathbb{R}$. But $f^{-1}$ is not differentiable in $0$.

💡 Let $f:D⟶\mathbb{R},D\subseteq \mathbb{R}$ and $x_0\in D$.

1. $f$ has a local maximum in $x_0$ if there exists $\delta >0$ with:

   $$f(x)\le f(x_0)\forall x\in (x_0-\delta ,x_0+\delta )\cap D$$

2. $f$ has a local minimum in $x_0$ if there exists $\delta >0$ with:

   $$f(x)\ge f(x_0)\forall x\in (x_0-\delta ,x_0+\delta )\cap D$$

3. $f$ has a local extremum in $x_0$ if $x_0$ is either a local minimum or maximum of $f$.

📖 Let $f:(a,b)⟶\mathbb{R}$, $x_0\in (a,b)$. We assume that $f$ is differentiable in $x_0$.

1. If $f^{\prime }(x_0)>0$ then there exists $\delta >0$ with

   $$\begin{array}{rlrl}& f(x)>f(x_0)& \forall x\in (x_0,x_0+\delta )& \\ & f(x)<f(x_0)& \forall x\in (x_0-\delta ,x_0)& .\end{array}$$

2. If $f^{\prime }(x_0)<0$ then there exists $\delta >0$ with

   $$\begin{array}{rlrl}& f(x)<f(x_0)& \forall x\in (x_0,x_0+\delta )& \\ & f(x)>f(x_0)& \forall x\in (x_0-\delta ,x_0)& .\end{array}$$

3. If $f$ has a local extremum in $x_0$, then $f^{\prime }(x_0)=0$.

📖 (Rolle 1690).

Let $f:[a,b]⟶\mathbb{R}$ be continuous and differentiable in $(a,b)$. If $f(a)=f(b)$ holds true, then there exists $\xi \in (a,b)$ with

$$f^{\prime }(\xi )=0.$$

📖 (Lagrange 1797).

Let $f:[a,b]⟶\mathbb{R}$ be continuous and differentiable in $(a,b)$. Then there exists $\xi \in (a,b)$ with

$$f(b)-f(a)=f^{\prime }(\xi )(b-a).$$

📎 Let $f,g:[a,b]⟶\mathbb{R}$ be continuous and differentiable in $(a,b)$.

1. If $f^{\prime }(\xi )=0\forall \xi \in (a,b)$ then $f$ is constant.
2. If $f^{\prime }(\xi )=g^{\prime }(\xi )\forall \xi \in (a,b)$ then there exists $c\in \mathbb{R}$ with $f(x)=g(x)+c\forall x\in [a,b]$.
3. If $f^{\prime }(\xi )\ge 0\forall \xi \in (a,b)$ then $f$ is monotonically increasing on $[a,b]$.
4. If $f^{\prime }(\xi )>0\forall \xi \in (a,b)$ then $f$ is strictly monotonically increasing on $[a,b]$.
5. If $f^{\prime }(\xi )\le 0\forall \xi \in (a,b)$ then $f$ is monotonically decreasing on $[a,b]$.
6. If $f^{\prime }(\xi )<0\forall \xi \in (a,b)$ then $f$ is strictly monotonically decreasing on $[a,b]$.

7. If there exists $M\ge 0$ with

   $$|f^{\prime }(\xi )|\le M\forall \xi \in (a,b)$$

   then it follows that $\forall x_1,x_2\in [a,b]$:

   $$|f(x_1)-f(x_2)|\le M|x_1-x_2|.$$

📖 (Cauchy).

Let $f,g:[a,b]⟶\mathbb{R}$ be continuous and differentiable in $(a,b)$. Then there exists $\xi \in (a.b)$ with

$$g^{\prime }(\xi )(f(b)-f(a))=f^{\prime }(\xi )(g(b)-g(a)).$$

If $g^{\prime }(x)\ne 0\forall x\in (a,b)$ then it follows that

$$g(a)\ne g(b)$$

and

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

📖 (l’Hospital 1696, Johann Bernoulli 1691/92).

Let $f,g:(a,b)⟶\mathbb{R}$ be differentiable with $g^{\prime }(x)\ne 0\forall x\in (a,b)$.

If

$$\underset{x\to b^{-}}{\lim }f(x)=0,\underset{x\to b^{-}}{\lim }g(x)=0$$

and

$$\underset{x\to b^{-}}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}=\lambda$$

exist, then

$$\underset{x\to b^{-}}{\lim }\frac{f(x)}{g(x)}=\underset{x\to b^{-}}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}.$$

💡 Let $I\subseteq \mathbb{R}$ be an interval and let $f:I⟶\mathbb{R}$ be a function

1. $f$ is convex (on $I$) if for all $x\le y,x,y\in I$ and $\lambda \in [0,1]$ it holds that

   $$f(\lambda x+(1-\lambda )y)\le \lambda f(x)+(1-\lambda )f(y)$$

2. $f$ is strictly convex if for all $x<y,x,y\in I$ and $\lambda \in (0,1)$ it holds that

   $$f(\lambda x+(1-\lambda )y)<\lambda f(x)+(1-\lambda )f(y)$$

💡 Let $f:I⟶\mathbb{R}$ be convex. Then the following holds for all $n\ge 1,\{x_1,\dots ,x_n\}\subseteq I$ and ${\lambda }_1,\dots ,{\lambda }_n$ in $[0,1]$ with ${\sum }_{i=1}^n{\lambda }_i=1$:

$$f(\sum _{i=1}^n{\lambda }_i{x}_i)\le \sum _{i=1}^n{\lambda }_{i}f(x_i).$$

📌 Let $f:I⟶\mathbb{R}$ be an arbitrary function. The function $f$ is convex if and only if for all $x_0<x<x_1$ in $I$ it holds that

$$\frac{f(x)-f(x_0)}{x-x_0}\le \frac{f(x_1)-f(x)}{x_1-x}.$$

$\ast f$ is strictly convex if*

$$\frac{f(x)-f(x_0)}{x-x_0}<\frac{f(x_1)-f(x)}{x_1-x}.$$

📖 Let $f:(a,b)⟶\mathbb{R}$ be differentiable in $(a,b)$. The function $f$ is (strictly) convex if and only if $f^{\prime }$ is (strictly) monotonically increasing.

💡 Let $f:(a,b)⟶\mathbb{R}$. If $f$ is differentiable in $(a,b)$ and its derivative $f^{\prime }$ is differentiable in $(a,b)$, then $f^{\prime \prime }$ (or $f^{(2)}$) denotes the function $(f^{\prime }{)}^{\prime }$. The function $f^{\prime \prime }$ is called the second derivative of $f$ and $f$ can be differentiated twice in $(a,b)$.

📎 Let $f:(a,b)⟶\mathbb{R}$ be a function such that it can be differentiated twice in $(a,b)$.Then the function $f$ is (strictly) convex if $f^{\prime \prime }\ge 0$ (respectively $f^{\prime \prime }>0$) on $(a,b)$.

💡 Let $D\subseteq \mathbb{R}$ such that every $x_0\in D$ is an accumulation point of $D$. Let $f:D⟶\mathbb{R}$ be differentiable in $D$ and let $f^{\prime }$ be its derivative, we set $f^{(1)}=f^{\prime }$.

1. For $n\ge 2$ $f$ can be differentiated $n$-times in $D$ if $f^{(n-1)}$ is differentiable in $D$. Then $f^{(n)}=(f^{(n-1)}{)}^{\prime }$ and is called the $n$-th derivative of $f$.
2. The function $f$ can be differentiated $n$-times continuously in $D$, if it can be differentiated $n$-times and if $f^{(n)}$ is continuous in $D$.
3. The function $f$ is smooth in $D$, if it can be differentiated $n$-times for all $n\ge 1$.

💡 It follows that for $n\ge 1$ a function that can be differentiated $n$-times can be differentiated $(n-1)$-times continuously.

📖 Let $D\subseteq \mathbb{R}$ such that every $x_0\in D$ is an accumulation point of $D$. Let $n\ge 1$ and $f,g:D⟶\mathbb{R}$ be differentiable $n$-times in $D$.

1. $f+g$ can be differentiated $n$-times and

   $$(f+g{)}^{(n)}=f^{(n)}+g^{(n)}.$$

2. $f\cdot g$ can be differentiated $n$-times and

$$(f\cdot g{)}^{(n)}=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})f^{(k)}{g}^{(n-k)}.$$

📖 Let $D\subseteq \mathbb{R}$ such that every $x_0\in D$ is an accumulation point of $D$. Let $n\ge 1$ and $f,g:D⟶\mathbb{R}$ be differentiable $n$-times in $D$.

If $g(x)\ne 0\forall x\in D$, then $\frac{f}{g}$ is differentiable $n$-times in $D$.

📖 Let $E,D\subseteq \mathbb{R}$ be subsets of the reals such that each of their points is an accumulation point for its respective set. Let $f:D⟶E$ and $g:E⟶\mathbb{R}$ be differentiable $n$-times. Then $g\circ f$ is differentiable $n$-times and

$$(g\circ f{)}^{(n)}(x)=\sum _{k=1}^n{A}_{n,k}(x)(g^{(k)}\circ f)(x)$$

where $A_{n,k}$ is a polynomial in the functions $f^{\prime },f^{(2)},\dots ,f^{(n+1-k)}$.