Differentiation of a determinant:
where a dash shows derivative with respect to x.
Example: Suppose a be a repeated root of the quadratic equation f(x) =0 and A(x), B(x), C(x) be polynomials of degree 3, 4, and 5 respectively. Then defines that
is divisible by f(x), where a dash shows the derivative, with respect to x.
Solution: Consider g(x) = 
We calculate that g(a)=0, & g'(a)=0
=> g(x)=(x-a)2 P(x), where P(x) is a polynomial of degree 3.
Also f(x)=0 is a quadratic equation obtaining repeated root a
=> g(x) is divisible by f(x).
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