If g(x)=(x−a)(x−b)(x−c)f(x), where f(x) is a polynomial of degree <3, then prove that dxdg(x)=∣∣111abcf(a)(x−a)−2f(b)(x−b)−2f(c)(x−c)−2∣∣÷∣∣a2b2c2abc111∣∣.

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Question

If $g (x) = \frac{f ( x )}{( x - a ) ( x - b ) ( x - c )},$ where f(x) is a polynomial of degree $< 3,$ then prove that
$\frac{d g ( x )}{d x} = ∣ ∣ 111 a b c f (a) (x - a)^{- 2} f (b) (x - b)^{- 2} f (c) (x - c)^{- 2} ∣ ∣ \div ∣ ∣ a^{2} b^{2} c^{2} a b c 111 ∣ ∣ .$

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Liam Discussed

g (x) = \frac{f ( x )}{( x - a ) ( x - b ) ( x - c )},

where f(x) is a polynomial of degree

< 3,

then prove that

\frac{d g ( x )}{d x} = ∣ ∣ 111 a b c f (a) (x - a)^{- 2} f (b) (x - b)^{- 2} f (c) (x - c)^{- 2} ∣ ∣ \div ∣ ∣ a^{2} b^{2} c^{2} a b c 111 ∣ ∣ .

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Question Text

g (x) = \frac{f ( x )}{( x - a ) ( x - b ) ( x - c )},

where f(x) is a polynomial of degree

< 3,

then prove that

\frac{d g ( x )}{d x} = ∣ ∣ 111 a b c f (a) (x - a)^{- 2} f (b) (x - b)^{- 2} f (c) (x - c)^{- 2} ∣ ∣ \div ∣ ∣ a^{2} b^{2} c^{2} a b c 111 ∣ ∣ .