From a4931ccf377daa616c3d1633281e43d0409c3055 Mon Sep 17 00:00:00 2001 From: zacharias Date: Wed, 19 Nov 2025 15:00:24 +0100 Subject: [PATCH] vault backup: 2025-11-19 15:00:24 --- .obsidian/workspace.json | 25 ++++++++++++++++++++++--- Derivata.md | 29 +++++++++++++++++++++++++++++ Pasted image 20251119134315.png | Bin 0 -> 8203 bytes d1.png | Bin 0 -> 34369 bytes d_ex_1.png | Bin 0 -> 11217 bytes 5 files changed, 51 insertions(+), 3 deletions(-) create mode 100644 Derivata.md create mode 100644 Pasted image 20251119134315.png create mode 100644 d1.png create mode 100644 d_ex_1.png diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 0f59ad9..97347b4 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -21,6 +21,20 @@ "title": "Gräsvärde (1)" } }, + { + "id": "ad6eb280b4b8718c", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Derivata.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Derivata" + } + }, { "id": "66704e0159322e3f", "type": "leaf", @@ -91,7 +105,8 @@ "title": "Funktioner Forts" } } - ] + ], + "currentTab": 1 } ], "direction": "vertical" @@ -252,12 +267,16 @@ "obsidian-git:Open Git source control": false } }, - "active": "5d5c0fef64eecc2b", + "active": "e616c86f78b96cf1", "lastOpenFiles": [ + "Pasted image 20251119134315.png", + "d_ex_1.png", + "d1.png", + "Gräsvärde (1).md", + "Derivata.md", "Funktioner.md", "Funktioner Forts.md", "Komplexa tal.md", - "Gräsvärde (1).md", "Trigonometri.md", "Grafer.md", "conflict-files-obsidian-git.md", diff --git a/Derivata.md b/Derivata.md new file mode 100644 index 0000000..23e8c4a --- /dev/null +++ b/Derivata.md @@ -0,0 +1,29 @@ +- Derivata + - **Def**: *$f$ är deriverbar i punkten $a$ om $$\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$existerar.$$f'(x)=\frac{df}{dx}(a)=Df(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$är derivatan av $f$ i punkten $x=a$. Funktionen $f'$ är derivatan av $f$ och deinieras som $x\longmapsto f'(x)$ där det är definiead.* + - **Defs**: + - $Df$: *Oendlig liten ändrig i $f$* + - $Dx$: *Oendlig liten ändrig i $x$* + - $f[\bullet]=f'$ + - +![[d1.png]] +- Egenskaper och regler + - $f$ deriverbar $\Rightarrow$ $f$ kontinuerlig. **Obs!** Inte alla kontinuerliga funktioner är deriverbara + - Derivering är linjär avbildning: $\left(\alpha f+\beta g\right)'=\alpha f'+\beta g'$ + - **Produkt regel** (*Leibniz*): $\left(f\left(x\right)g\left(x\right)\right)'=f'\left(x\right)g\left(x\right)+f\left(x\right)g'\left(x\right)$ + - **Sammansatt funktion**: $\left(f\circ g\right)'\left(x\right)=f'\circ g\left(x\right)g'\left(x\right)$ + - **Division**: $\left(\frac{f\left(x\right)}{g\left(x\right)}\right)'=\frac{f'\left(x\right)g\left(x\right)-f\left(x\right)g'\left(x\right)}{g\left(x\right)^2}$ + - **Ex**: ![[d_ex_1.png]]$$\begin{align}f(x)=\mid x\mid\\f\text{ är kontinuerlig på }\mathbb{R}.\\f\text{ är inte deriverbar i }0.\\\lim_{x\to0+}\frac{f\left(x\right)-f\left(0\right)}{x-0}=\lim_{x\to0+}\frac{\mid x\mid-0}x=\lim_{x\to0+}\frac xx=1\\\lim_{x\to0-}\frac{f\left(x\right)-f\left(0\right)}{x-0}=\lim_{x\to0-}\frac{\mid x\mid-0}x=\lim_{x\to0-}\frac{-x}x=-1\\\lim_{x\to0}\frac{f\left(x\right)-f\left(0\right)}{x-0}=f'(0)\text{ existerar inte-}\end{align}$$ + - **Ex**: $$\begin{align}\text{Leibniz regel}\\\left(f\left(x\right)g\left(x\right)\right)'=\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)g\left(x\right)}h\\=\lim_{h\to0}\frac{f(x+h)g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)}{h}\\=\lim_{h\to0}\left(g(x+h)\frac{f(x+h)-f(x)}{h}+f(x)\frac{g(x+h)-g(x)}{h}\right)\\=g(x)f'(x)+f(x)g'(x)\end{align}$$ + - **Ex**: $$\begin{align}h(x)=\frac1x\\h'(x)=-\frac1{x^2}\\h\circ g(x)=h(g(x))=\frac1{g(x)}\\(g\circ g)'(x)=\left(\frac1{g(x)}\right)^2\\=h'\circ g(x)g'(x)=h'(g(x))g'(x)\frac{-1}{(g(x))^2}g'(x)\end{align}$$ +- Standerd derivarives + 1. $f(x)=c\;\;\Rightarrow\;\;f'(x)=0$ + 2. $f(x)=n^n\;\;\Rightarrow\;\;f'(x)=nx^{n-1},\;n\in\mathbb{Z}$ + 3. $f(x)=x^\alpha\;\;\Rightarrow\;\;f'(x)=\alpha x^{\alpha-1},\;\alpha\in\mathbb{R},\;x>0$ + 4. $f(x)=e^x\;\;\Rightarrow\;\;f'(x)=e^x$ + 5. $f(x)=\ln\mid x\mid\;\;\Rightarrow\;\;f'(x)=x^{-1},\;x\neq0$ + 6. $f(x)=\sin x\;\;\Rightarrow\;\;f'(x)=\cos x$ + 7. $f(x)=\cos x\;\;\Rightarrow\;\;f'(x)=-\sin x$ + 8. $f(x)=\tan x\;\;\Rightarrow\;\;f'(x)=\sec^2x=1+\tan^2x$ + 9. $f(x)=a^x\;\;\Rightarrow\;\;f'(x)=a^x\ln a,\;a>0$ + 10. $f(x)=\log_a\mid x\mid\;\;\Rightarrow\;\;f'(x)=(x\ln a)^-1,\;a>0,\;x\neq0$ + 11. \ No newline at end of file diff --git a/Pasted image 20251119134315.png b/Pasted image 20251119134315.png new file mode 100644 index 0000000000000000000000000000000000000000..077bf8fd2492ea400badf84945f89bd72a867010 GIT binary patch literal 8203 zcmeAS@N?(olHy`uVBq!ia0y~yVEoO%z^Ki^#K6E1c75+E1_lPs0*}aI1_r((Aj~*b zn@^g7L4L2Ni(^Q|oVR!D3u3-jzW?aHsV+Kdz3zXelh;ph+}@;I*2V5# 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