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10.31 - Last time we saw one way to compute the derivative...

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Last time we saw one way to compute the derivative of x x with respect to x . Another method we can apply is logarithmic differentiation: ( d / dx ) ln f ( x ) = f ( x ) / f ( x ) That is, if f ( x ) > 0 on some open interval I , then ln f ( x ) is differentiable with derivative f ( x )/ f ( x ) throughout I . (Proof: apply the chain rule.) The consequence f ( x ) = f ( x ) ( d / dx ) ln f ( x ) is frequently useful. Example: Let f ( x ) = x x (for x > 0). Then ln f ( x ) = x ln x and ( d / dx ) ln f ( x ) = ( x )(1/ x ) + (1)(ln x ) = 1 + ln x so f ( x ) = f ( x ) ( d / dx ) ln f ( x ) = ( x x ) (1 + ln x ) as derived above. Questions? General guideline: when functions involve logs and
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exponentials, simplify algebraically before you differentiate; you’ll save yourself lots of trouble! Section 3.4: Exponential growth and decay Let f ( t ) = # of bacteria in a colony at time t (where t is measured in seconds) Bacteria reproduce, so the colony will grow, with f ( t ) > 0 Assume the colony grows, with no bacteria dying Then f ( t ) = rate at which new bacteria are added to the colony It’s reasonable to assume that until the colony starts to run out of resources (food, growing-space), f ( t ) will be proportional to f ( t ). That is, f ( t ) = k f ( t ) for some constant k > 0. Let g ( t ) = # of atoms of plutonium in a sample at time t Plutonium decays, so the sample will shrink, with g ( t ) < 0 g ( t ) = rate at which atoms in the sample decay g ( t ) = k g ( t ) for some constant k > 0.
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