# Let's do some calculus! (11)

Calculus Level 4

Let $$f_{r}(x), g_{r}(x), h_{r}(x)$$ be polynomials defined for $$r \in \{1,2,3\}$$ such that $$f_{r}(a)=g_{r}(a)=h_{r}(a)$$ for each and every $$r \in \{1,2,3\}$$.

Let another polynomial $$F(x)$$ be defined as:

$F(x) = \begin{vmatrix} f_{1}(x) & f_{2}(x) & f_{3}(x) \\ g_{1}(x) & g_{2}(x) & g_{3}(x) \\ h_{1}(x) & h_{2}(x) & h_{3}(x) \end{vmatrix}$

Find $$F'(a)$$.

Notations: $$F'(x)$$ denotes the first derivative of $$F(x)$$.

Clarification:

Each and every $$r \in \{1,2,3\}$$ means that for any $$r$$ chosen from the set $$\{1,2,3\}$$, $$f_{r}(a)=g_{r}(a)=h_{r}(a)$$. Such as, $$f_{1}(a)=g_{1}(a)=h_{1}(a)$$, $$f_{2}(a)=g_{2}(a)=h_{2}(a)$$ and $$f_{3}(a)=g_{3}(a)=h_{3}(a)$$ respectively.