f ( 2 ) = 5
f ' ( 2 ) = -0,5
Ableitung e-Funktion
( e ^term ) ´ = e ^term * ( term ´ )
f ( x ) = a * e ^{b*x}
term = b * x
term ´ = b
f ´( x ) = a * e ^{b*x} * b
f ( 2 ) = a * e ^{b*2} = 5
f ´( 2 ) = a * e ^{b*2} * b = -0.5
a * e ^{b*2} = 5
a * e ^{b*2} * b = -0.5 | teilen
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1 / b = 5 / -0.5
b = -0.5 / 5
b = -0.1
Einsetzen
a * e ^{-0.1*2} = 5
a * e ^{-0.2} = 5
a = 5 / e ^{-0.2}
a = 6.11
Probe
6.11 * e ^{-0.1*2} * (-0.1) = -0.5
Stimmt