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i did a thing

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Christoph J. Scherr 2024-09-05 14:47:57 +02:00
parent 151f8ebb19
commit 79d78609f1
2 changed files with 13 additions and 0 deletions
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src/exercise/1.typ
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@ -109,5 +109,18 @@ $
#set enum(numbering: "(a)")
+ $
e(x) = e^x wide a(x) &= (e^x)/(1+e^x) wide b(x) = 1+e^x \
ln: RR^+ |-> RR wide e&: RR |-> RR^+ wide => f: RR |-> RR <=> DD_f = RR \
e'(x) = e(x) wide ln'(x) &= 1/x wide b'(x) = e^x \
a'(x) &= (b dot e' - b' dot e)/(b^2) space #text("Quotientenregel") \
a'(x) &= ([1+e^x] dot e^x - e^x dot e^x)/([1+e^x]^2) \
a'(x) &= (e^x + e^(2x) - e^(2x))/(1+2e^x+e^(2x)) = (e^x)/([1+e^x]^2) \
f'(x) &= ln'(a(x)) dot a'(x) \
&= 1/((e^x)/(1+e^x)) dot (e^x)/([1+e^x]^2) \
&= (1+e^x)/(e^x) dot (e^x)/([1+e^x]^2) \
&= (1+e^x)/([1+e^x]^2) \
&= 1/(1+e^x) checkmark
$