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Interesting differentiation Problem

Let \(y = 1 + \frac {a_1}{x- a_1} + \frac {a_2}{(x-a_1)(x-a_2)} + \frac {a_3}{(x-a_1)(x-a_2)(x-a_3)}+ \ldots + \frac {a_n}{(x-a_1)(x-a_2) \ldots (x-a_n)} \)

Prove that:

\(\frac {dy}{dx} = \frac {y}{x} ( \frac {a_1}{a_1-x} + \frac {a_2}{a_2 - x} + \ldots + \frac{a_n }{a_n - x} )\).

\(a_i\)are constants.

Hint : check for small values of \(n\) first, and then it should be done.

Note by Aditya Parson
4 years, 2 months ago

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This does not look right. Setting \(a_1 = a_2 = \dots = a_{n-1} = 0\) and \(a_n = 1\) seems to give different results from what you claim.

Peiyush Jain - 4 years, 2 months ago

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First add first two terms, then the result with the 3rd term & so on... At what you get, apply log & differentiate... And my boy,you will GET your result!! [Source: This is a very reputed problem in J.E.E.]

Paramjit Singh - 4 years, 2 months ago

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