Chapter 1, Problem 8

[dstefan]

Chapter 1, Problem 8:

Let f(y) = \sum_{i=1}^n c_i e^{-y t_i}, where c_i and t_i, i=1:n, are positive constants. Compute f'(y) and f''(y).


Solution:

Note that
\left( e^{-y t_i} \right)^{'} ~=~ \frac{d}{dy} \left( e^{-y t_i} \right) ~=~ - t_i e^{-y t_i}
\left( e^{-y t_i} \right)^{''} ~=~ \frac{d}{dy} \left( - t_i e^{-y t_i} \right) ~=~ t_i^2 e^{-y t_i}


Then,
f'(y) ~=~ - \sum_{i=1}^n c_i t_i e^{-y t_i}
f^{''}(y) ~=~ \sum_{i=1}^n c_i t_i^2 e^{-y t_i}