If `g(x)=(f(x))/((x-a)(x-b)(x-c)),` where f(x) is a polynomial of degree `lt3,` then prove that
`(dg(x))/(dx)=|{:(1,a,f(a)(x-a)^(-2)),(1,b,f(b)(x-b)^(-2)),(1,c,f(c)(x-c)^(-2)):} |divide|{:(a^(2),a,1),(b^(2),b,1),(c^(2),c,1):}|.`

Explanation

By partial fractions we have
`g(x)=(f(a))/((x-a)(a-b)(a-c))+(f(b))/((b-a)(x-b)(b-c))`
`+(f(c ))/((c-a)(c-b)(x-c))`
`=(1)/((a-b)(b-c)(c-a))xx`
`[(f(a)(c-b))/((x-a))+(f(b)(a-c))/((x-b))+(f(c )(b-a))/((x-c))]`
`= |{:(1,,a,,f(a)//(x-a)),(1,,b,,f(b)//(x-b)),(1,,c,,f(c )//(x-c)):}|-:|{:(1,,a,,a^(2)),(1,,b,,b^(2)),(1,,c,,c^(2)):}|`
`rArr (dg(x))/(dx)= |{:(1,,a,,-f(a)(x-a)^(-2)),(1,,b,,-f(b)(x-b)^(-2)),(1,,c,,-f(c )(x-c)^(-2)):}|-:|{:(1,,a,,a^(2)),(1,,b,,b^(2)),(1,,c,,c^(2)):}|`
`= |{:(1,,a,,f(a)(x-a)^(-2)),(1,,b,,f(b)(x-b)^(-2)),(1,,c,,f(c )(x-c)^(2)):}|-:|{:(a^(2),,a,,1),(b^(2),,b,,1),(c^(2),,c,,1):}|`