As an alternative proof that the functions are linearly independent on (∞,-∞) when are distinct, assume holds for all x in (∞,-∞) and proceed as follows:
(a) Because the ri’s are distinct we can (if necessary)relabel them so that .Divide equation (33) by to obtain Now let x→∞ on the left-hand side to obtain C1 = 0. (b) Since C1 = 0, equation (33) becomes
= 0for all x in(∞,-∞). Divide this equation by
and let x→∞ to conclude that C2 = 0.
(c) Continuing in the manner of (b), argue that all thecoefficients, C1, C2, . . . ,Cn are zero and hence are linearly independent on (∞,-∞).
The answer to this problem is:
The Sum rule says the derivative of a sum of functions is the sum of their derivatives. The Difference rule says the derivative of a difference of functions is the difference of their derivatives.
We will do the following question on the basis of basic differentiation ;
Hence, the final answer is:
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