# 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.