derivative question

[匿名]

2010-07-18 10:37 am

P(t) = ( (K) (L) ) / (L + ( K- L) e^(-r t) )

both L and r are constant

SHOW that the derivative of P (t) is

P ' (t) = r ( (KL) / ( L+ (K-L) e^ (-r t ) ) times ( 1- ( L / ( L + ( K - L ) e^ (-r t ) ) ) )

Please explain in details, thank you very much!!!!

回答 (1)

六呎將軍

2010-07-18 5:14 pm

✔ 最佳答案

With P(t) = KL/[L + (K - L)e-rt], we let u = L + (K - L)e-rt and then P(t) = KL/u

Applying the Chain Rule:

P(t) = KLu-1

P(t) = -KLu-2 xu/dt

= -KL/[L + (K - L)e-rt]2 x d[L + (K - L)e-rt]/dt

= -KL/[L + (K - L)e-rt]2 x [-r(K - L)e-rt]

= rKL(K - L)e-rt/[L + (K - L)e-rt]2

= {rKL[L + (K - L)e-rt] - rKL2}/[L + (K - L)e-rt]2

= rKL{[L + (K - L)e-rt] - L}/[L + (K - L)e-rt]2

= {rKL/[L + (K - L)e-rt]} x {[L + (K - L)e-rt] - L}/[L + (K - L)e-rt]

= {rKL/[L + (K - L)e-rt]} x {1 - L/[L + (K - L)e-rt]}

參考: Myself

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