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.

#Calculus #MathProblem #Math #Opinions

Easy Math Editor

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`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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Comments

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 - 7 years, 10 months ago

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.]

A Brilliant Member - 7 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$