Mathe am 18.11.2024

schule/mathe/MA_2024-11-18.typ

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+#import "@preview/grape-suite:1.0.0": exercise
+#import exercise: project, task, subtask
+
+#set text(lang: "de")
+
+#show: project.with(
+  title: [Kettenregel],
+  seminar: [Mathe Q2],
+  show-outline: true,
+  author: "Erik Grobecker",
+  date: datetime(day: 18, month: 11, year: 2024),
+  show-solutions: false,
+)
+
+#show heading.where(level: 1): it => {
+  counter(math.equation).update(0)
+  it
+}
+
+#show math.equation: set text(font: "New Computer Modern Math")
+
+= Kettenregel
+
+$
+  f(x)&= underbrace((3/4 x^2 -3), u)  dot underbrace(e^(1.4-x^2), v) \
+  f'(x)&=u' dot v + u dot v'\
+
+  u'&=3/4 dot 2x^1\ &
+  = 3/2 dot x\
+
+  v'&=e^(1.4-x^2)\
+  &=e^(overbrace(1.4-x^2, "innere Ableitung")) \
+  &=e^(1.4-x^2) dot overbrace((-2x), "innere Ableitung")\
+  \
+  "Einsetzen:"\
+  f'(x)&=u' dot v + u dot v'\
+  &=3/2x dot e^(1.4-x^2) + (3/4 x^2 -3) dot e^(1.4-x^2) dot (-2x) #h(2em)&| e "entfernen" \
+  &=e^(1.4-x^2) dot (3/2x + (3/4x^2-3) dot (-2x))\
+  &=e^(1.4-x^2) dot (3/2x -3/2x^3 + 6x)\
+  &=e^(1.4-x^2) dot (-3/2x^3 + 15/2x)
+$
+
+Innere Ableitung:
+$
+  u(x)&=1.4-x^2\
+  u'(x)&=2x
+$
+
+#pagebreak()
+
+== Aufgabe
+Leite die folgenden Funktionen ab:
+
+*1)*
+$
+  f(x)&=e^(2x)\
+  f'(x)&=e^(2x)dot 2\
+  &=2e^(2x)
+$
+
+*2)*
+$
+  f(x)&=e^(3x)\
+  f'(x)&=e^(3x) dot 3\
+  &=3e^(3x)
+$
+
+*3)*
+$
+  f(x)&=e^(-x)\
+  f'(x)&=e^(-x) dot (-1)\
+  &=-e^(-x)
+$
+
+*4)*
+$
+  f(x)&=e^(0.5x)\
+  f'(x)&=e^(0.5x) dot 1 / 2\
+  &=1 / 2 e^(0.5x)
+$