Derivative

Differentiability

given $x_0\in I\subseteq \text{dom}(f)$

• $f$ is differentiable at a point $x_0\in I$
• (7.8) The limit $L=\underset{h\to 0}{\lim }\frac{f(x_0+h)-f(x_0)}{h}=\underset{x\to x_0}{\lim }\frac{f(x)-f(x_0)}{x-x_0}$ exists
• $\forall \epsilon >0,\exists \delta >0:\forall h\left(0<|h|<\delta ⟹\left|L-\frac{f(x_0+h)-f(x_0)}{h}\right|<\epsilon \right)$
• $f$ is left- and right-differentiable at $x_0$ and $f_{+}^{\prime }(x_0)=f_{-}^{\prime }(x_0)$ (in this case $f^{\prime }(x_0)=f_{+}^{\prime }(x_0)=f_{-}^{\prime }(x_0)$)
• todo $\exists A\in \mathbb{R}:f(x_0+h)-f(x_0)=Ah+\alpha (h)h$ where $\underset{h\to 0}{\lim }\alpha (h)=0$
• There exists a function $g$ defined on $I$ and continuous at $x_0$ s.t. $\forall x\in I,f(x)=g(x)\cdot (x-x_0)+f(x_0)$

Derivative

• Derivative Function - $f^{\prime }(x)=\frac{df}{dx}=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$f(x)$ is differentiable at a point $x_0\in \text{dom}(f)$

• Derivative of $f$ at the point $x_0$ is $f^{\prime }(x_0)$
  • $f^{\prime }(x_0)=\underset{h\to 0}{\lim }\frac{f(x_0+h)-f(x_0)}{h}=\underset{x\to x_0}{\lim }\frac{f(x)-f(x_0)}{x-x_0}$
  • $f^{\prime }(x_0)$ is the slope of the tangent line to the graph of $f$ at the point $x_0$
  • $f^{\prime }(x_0)$ is the instantaneous rate of change of $f$ with respect to $x$ at $x_0$

Notation \displaystyle f'(x_{0})=\frac{df}{dx}(x_{0}) = \left.{\frac {df}{dx}}\right|_{x=x_{0}}

Rate of Change

• The average rate of change of $y=f(x)$ with respect to $x$ over the interval $[x_0,x_1]$ is $\frac{\Delta y}{\Delta x}=\frac{f(x_1)-f(x_0)}{x_1-x_0}=\frac{f(x_0+h)-f(x_0)}{h}$ (where $h\ne 0$)
• The instantaneous rate of change of $f$ with respect to $x$ at $x_0$ is the derivative $f^{\prime }(x_0)=\underset{h\to 0}{\lim }\frac{f(x_0+h)-f(x_0)}{h}$. (provided the limit exists)

Theorems

• if $f$ is differentiable at $x_0$

  • (7.9) $f$ continuous at $x_0$
  • (q7.62) There exists a neighborhood on which $f$ is bounded

• if $f$ and $g$ are differentiable at $x_0$

  • (q7.19) Piecewise Function $k(x)=\{\begin{array}{ll}f(x)& x\le x_0\\ g(x)& x>x_0\end{array}$
    • $k$ is continuous at $x_0$ $⟺f(x_0)=g(x_0)$
    • $k$ is differentiable at $x_0$ $⟺f(x_0)=g(x_0)$ and $f^{\prime }(x_0)=g^{\prime }(x_0)$
    • https://en.wikipedia.org/wiki/Piecewise#Continuity_and_differentiability_of_piecewise-defined_functionstodo

• (Local Property) $\exists \delta :\forall x\in (x_0-\delta ,x_0+\delta ),f(x)=g(x)⟹f^{\prime }(x_0)=g^{\prime }(x_0)$

• (q8.24) if $f$ is continuous at $x_0$, and $f$ differentiable at deleted neighborhood of $x_0$, and $\underset{x\to x_0}{\lim }{f}^{\prime }(x)\in \mathbb{R}$ then $f$ differentiable at $x_0$ and $f^{\prime }(x_0)=\underset{x\to x_0}{\lim }{f}^{\prime }(x)$

Derivative Rules

Assuming the function are differentiable at the relevant points

Derivative Rules
Sum/Difference Rule	$(f\pm g{)}^{\prime }(x_0)=f^{\prime }(x_0)\pm g^{\prime }(x_0)$
Product Rule (Leibniz rule)	$(fg{)}^{\prime }=f{g}^{\prime }+f^{\prime }g$
---- Constant Multiple Rule	$(cf{)}^{\prime }(x_0)=c\cdot f^{\prime }(x_0)$
---- (7.17)	$(f^n{)}^{\prime }(x)=n{f}^{n-1}(x)\cdot f^{\prime }(x)$
---- Power Rule	$(x^n{)}^{\prime }=n{x}^{n-1}$ (integer $n\ne 0$, 7.20) $(x^r{)}^{\prime }=r{x}^{r-1}$ ($r\in \mathbb{R}$, $x>0$, 7.31)
Quotient Rule	${\left(\frac{f}{g}\right)}^{\prime }(x_0)=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$	$g(x_0)\ne 0$
---- (7.19) Reciprocal rule	${\left(\frac{1}{f}\right)}^{\prime }=\frac{-f^{\prime }}{f^2}$ $\frac{d}{dx}\left[\frac{1}{f(x)}\right]=\frac{-f^{\prime }(x)}{[f(x){]}^2}$	$f(x_0)\ne 0$
Chain Rule	$(f(g(x)){)}^{\prime }=f^{\prime }(g(x))\cdot g^{\prime }(x)$	$\frac{dz}{dx}=\frac{dz}{dy}\cdot \frac{dy}{dx}$
	$(f(g(h(x))){)}^{\prime }=f^{\prime }(g(h(x)))\cdot g^{\prime }(h(x))\cdot h^{\prime }(x)$
	$f_{1..n}^{\prime }(x)=\prod _{k=1}^n{f}_k^{\prime }\left(f_{(k+1..n)}(x)\right)$
linearity of differentiation	$\frac{\text{mbox}d}{\text{mbox}dx}(\alpha \cdot f(x)+\beta \cdot g(x))=\alpha \cdot f^{\prime }(x)+\beta \cdot g^{\prime }(x)$	from the sum and constant factor rules

One Side

• $f$ is right-differentiable at $x_0$ if $L=\underset{h\to 0^{+}}{\lim }\frac{f(x_0+h)-f(x_0)}{h}$ exists.
  • (in that case $f_{+}^{\prime }(x_0)=L$ is the right-derivative of $f$ at $x_0$)
• $f$ is left-differentiable at $x_0$ if $L=\underset{h\to 0^{-}}{\lim }\frac{f(x_0+h)-f(x_0)}{h}$ exists.
  • (in that case $f_{-}^{\prime }(x_0)=L$ is the left-derivative of $f$ at $x_0$)

Differentiability on Interval

• $f$ is said to be differentiable on the interval $I$ if $f$ is differentiable at every point $x\in I\subseteq \text{dom}(f)$

Theorems

• $\text{dom}(f^{\prime })\subseteq \text{dom}(f)$

• Mean Value Theorem - $f$ (and $g$) is continuous on $[a,b]$, differentiable on the $(a,b)$

  • (8.5) (Rolle’s Theorem) - $f(a)=f(b)⟹\exists c\in (a,b):f^{\prime }(c)=0$
    • Rolle’s theorem is special case of MVT when $f(a)=f(b)$
  • (8.6) (Lagrange’s MVT) - $\exists c\in (a,b):f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$
    • MVT is special case of Cauchy’s MVT when $g(x)=x$
  • (8.9) (Cauchy’s MVT) if $\forall x\in (a,b),g^{\prime }(x)\ne 0$, then:
    • $g(a)\ne g(b)$ and $\exists c\in (a,b):\frac{f^{\prime }(c)}{g^{\prime }(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$

• $f$ is continuous on $I$, differentiable at every inferior point of $I$

  • First Derivative Test for Monotonicity (q8.28, 8.18, q8.29)
    • $f^{\prime }(x)\ge 0⟺$ $f$ is weakly increasing on $I$
    • $f^{\prime }(x)>0⟹$ $f$ is increasing on $I$
    • $f^{\prime }(x)\le 0⟺$ $f$ is weakly decreasing on $I$
    • $f^{\prime }(x)<0⟹$ $f$ is decreasing on $I$
    • $f$ is increasing on $I$, if and only if, $f^{\prime }(x)\ge 0$, and there is no subinterval on which $f^{\prime }(x)=0$ for all $x$ in it
    • $f$ is decreasing on $I$, if and only if, $f^{\prime }(x)\le 0$, and there is no subinterval on which $f^{\prime }(x)=0$ for all $x$ in it

• if $f$ is continuous on $[a,b]$ and monotonic on $(a,b)$ then $f$ is monotonic on $[a,b]$ (by q8.25)

• Constant function theorem

  • (q7.12b) $f\text{ is constant on dom}(f)⟹\forall x\in \text{dom}(f),f^{\prime }(x)=0$
  • (8.7) $f$ is continuous on $[a,b]$ and differentiable $(a,b)$ then:
    • $\forall x\in (a,b),f^{\prime }(x)=0⟹f\text{ is constant on }[a,b]$
  • $f$ is differentiable on $I=(a,b),(a,\infty ),(-\infty ,b),(-\infty ,\infty )$
    • (8.8, q8.11) $\forall x\in I,\left(f^{\prime }(x)=0⟹f(x)=c\right)$

• $f$ is continuous on $[0,\infty )$, differentiable on the $(0,\infty )$

  • $(\exists c>0:\forall x>0,f^{\prime }(x)\ge c)⟹\underset{x\to \infty }{\lim }f(x)=\infty$ (e2019c.91.q2b)

• $f$ is differentiable on $[a,b]$

  • (8.10) Darboux’s theorem - $f^{\prime }$ has the intermediate value property, i.e. $(\min (f^{\prime }(a),f^{\prime }(b))\le t\le \max (f^{\prime }(a),f^{\prime }(b)))⟹\exists c\in [a,b]:f^{\prime }(c)=t$
  • (q8.17) If $f^{\prime }(a)=f^{\prime }(b)=0$ and $\forall x\in (a,b),f^{\prime }(x)\ne 0$, then $\forall x\in (a,b),f^{\prime }(x)>0$ or $\forall x\in (a,b),f^{\prime }(x)<0$

• (8.11)todo

• Constant Difference Theoremnot-in-course If $f$ and $g$ are differentiable on an interval, and if $f^{\prime }(x)=g^{\prime }(x)$ for all $x$ in that interval, then $f-g$ is constant on the interval; that is, there is a constant k such that $f(x)-g(x)=k$ or, equivalently, $f(x)=g(x)+k$ for all $x$ in the interval.

Examples

$f(x)$	$\text{dom}(f)$	$f^{\prime }(x)$	$\text{dom}(f^{\prime })$
$\sqrt{x}$	$[0,\infty )$	$\frac{1}{2\sqrt{x}}$	$(0,\infty )$
$\sqrt{g(x)}$	$\{x\mid g(x)\ge 0\}$	$\frac{g^{\prime }(x)}{2\sqrt{g(x)}}$	$\{x\mid g(x)>0\}$
$\sin (x)$	$\mathbb{R}$	$\cos (x)$	$\mathbb{R}$
$\cos (x)$	$\mathbb{R}$	$-\sin (x)$	$\mathbb{R}$
$\tan (x)$	$\mathbb{R}\setminus \left\{\frac{(2n+1)\pi }{2}\mid n\in \mathbb{Z}\right\}$	$\frac{1}{{\cos }^{2}x}=1+{\tan }^2(x)$	$\text{dom}(\tan (x))$
$\cot (x)$	$\mathbb{R}\setminus \left\{n\pi \mid n\in \mathbb{Z}\right\}$	$-\frac{1}{{\sin }^2(x)}$	$\text{dom}(\cot (x))$
$\arcsin (x)$	$[-1,1]$	$\frac{1}{\sqrt{1-x^2}}$	$(-1,1)$
$\arccos (x)$	$[-1,1]$	$-\frac{1}{\sqrt{1-x^2}}$	$(-1,1)$
$\arctan (x)$	$\mathbb{R}$	$\frac{1}{1+x^2}$	$\mathbb{R}$
$\text{arccot}(x)$	$\mathbb{R}$	$\frac{-1}{1+x^2}$	$\mathbb{R}$
$a^x$	$\mathbb{R}$	$a^x\cdot \ln (a)$	$\mathbb{R}$
$e^x$	$\mathbb{R}$	$e^x$	$\mathbb{R}$
${\log }_a(x)$	$(0,\infty )$	$\frac{1}{x\ln (a)}$	$(0,\infty )$
$\ln (x)$	$(0,\infty )$	$\frac{1}{x}$	$(0,\infty )$

Inverse functions

assuming: $f$ is continuous and monotonic on $I$

• (7.27) $(f^{-1}{)}^{\prime }(x)=\frac{1}{f^{\prime }(f^{-1}(x))}$

• if $f^{\prime }(x_0)=0$ then $(f^{-1}{)}^{\prime }(f(x_0))$ is undefined

Concavity

todonot-in-course

• The graph of a differentiable function $f(x)$ is
  • concave up on an open interval $I$ if $f^{\prime }$ is increasing on $I$
  • concave down on an open interval $I$ if $f^{\prime }$ is decreasing on $I$
• A point $(x_0,f(x_0))$ where the graph of a function has a tangent line and where the concavity changes is an inflection point (or point of inflection)
• The Second Derivative Test for Concavitytodo
• At an inflection point $(x_0,f(x_0))$, either $f^{\prime \prime }(x_0)=0$ or $f^{\prime \prime }(x_0)$ fails to exist
• An inflection point $x_0$ can be categorized by $f^{\prime }$
  • if $f^{\prime }(x_0)=0$, the point is a stationary inflection point
  • if $f^{\prime }(x_0)\ne 0$, the point is a non-stationary inflection point
• A stationary inflection point is not a local extremum point

Higher-order derivatives

• A function $f$ is twice differentiable if and both $f^{\prime }$ and $(f^{\prime }{)}^{\prime }$ exist. In such case $(f^{\prime }{)}^{\prime }=f^{\prime \prime }$ is called the second derivative of $f$
• A function $f$ is n-times differentiable if $f^{\prime },f^{\prime \prime },f^{\prime \prime \prime }=f^{(3)},\dots ,f^{(n)}$ all exist. In such case $f^{(n)}$ is called the $n$-th derivative of $f$
• $f^{(n+1)}(x)=(f^{(n)}{)}^{\prime }(x)$ for every $x$ in which $f^{(n+1)}$ is defined
• $f=f^{(0)}$
• for infinite times see smooth function

Derivative Rules

• (IN2-4.1) if $f$ (and $g$) n-times differentiable functions in an interval $D$ then:
  • $(f\pm g)$ is n-times differentiable in $D$ and $(f\pm g{)}^{(n)}=f^{(n)}\pm g^{(n)}$
  • $(f\cdot g)$ is n-times differentiable in $D$ and for all $x$ in $D$ we have
    • (General Leibniz rule) $(f\cdot g{)}^{(n)}(x)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})f^{(n-k)}(x)\cdot g^{(k)}(x)$
    • $(c\cdot f{)}^{(n)}=c\cdot f^{(n)}$
• if $f_1,\dots f_m$ n-times differentiable in an interval $D$ then $f_1+\cdots +f_m$ is n-times differentiable in $D$ and $(f_1+\cdots +f_m{)}^{(n)}=f_1^{(n)}+\cdots +f_m^{(n)}$
• if $f(x)=x^n$ then
  • if $k<n$ then $f^{(k)}(x)=n\cdot (n-1)\cdots (n-k+1)x^{n-k}$
  • if $k=n$ then $f^{(k)}(x)\equiv n!$
  • if $k>n$ then $f^{(k)}(x)\equiv 0$

Continuously Differentiability

• A function $f$ is said to be of class $C^0$ if it is continuous (the class $C^0$ consists of all continuous functions)
• A function $f$ is said to be continuously differentiable and of class $C^1$ if the derivative $f^{\prime }(x)$ exists and is itself a continuous function.
• A function $f$ is of class $C^2$ if the first and second derivative of the function both exist and are continuous.
• A function $f$ is said to be of class $C^k$ if the first $k$ derivatives $f^{\prime }(x),f^{\prime \prime }(x),\dots ,f^{(k)}(x)$ all exist and are continuous.
• A function $f$ is to be smooth (or infinitely differentiable) and of class $C^{\infty }$ if it has derivatives of all orders (so all these derivatives are continuous)

The classes $C^k$ can be defined recursively by declaring $C^0$ to be the set of all continuous functions, and declaring $C^k$ for any positive integer $k$ to be the set of all differentiable functions whose derivative is in $C^{k-1}$. In particular, $C^k\subset C^{k-1}$ for every $k>0$. The class $C^{\infty }$ of smooth functions, is the intersection of the classes $C^k$ as $k$ varies over the non-negative integers.

Although the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity