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Question

If $$ \displaystyle f\left( x \right) =\begin{vmatrix}
2x & x^{2} & x^{3}\\
x^{2}+2x &1  & 3x+1\\
2x & 1-3x^{2} &5x
\end{vmatrix}$$ then find $${ f  }^{ ' }\left( 1 \right)$$.


Solution

$$f\left( x \right) =\begin{vmatrix} 2x & x^{ 2 } & x^{ 3 } \\ x^{ 2 }+2x & 1 & 3x+1 \\ 2x & 1-3x^{ 2 } & 5x \end{vmatrix}\\ \Rightarrow f'\left( x \right) =\begin{vmatrix} 2 & x^{ 2 } & x^{ 3 } \\ 2x+2 & 1 & 3x+1 \\ 2 & 1-3x^{ 2 } & 5x \end{vmatrix}+\begin{vmatrix} 2x & 2x & x^{ 3 } \\ x^{ 2 }+2x & 0 & 3x+1 \\ 2x & -6x & 5x \end{vmatrix}+\begin{vmatrix} 2x & x^{ 2 } & 3{ x }^{ 2 } \\ x^{ 2 }+2x & 1 & 3 \\ 2x & 1-3x^{ 2 } & 5 \end{vmatrix}\\ \Rightarrow f'\left( 1 \right) =\begin{vmatrix} 2 & 1 & 1 \\ 4 & 1 & 4 \\ 2 & -2 & 5 \end{vmatrix}+\begin{vmatrix} 2 & 2 & 1 \\ 3 & 0 & 4 \\ 2 & -6 & 5 \end{vmatrix}+\begin{vmatrix} 2 & 1 & 3 \\ 3 & 1 & 3 \\ 2 & -2 & 5 \end{vmatrix}=9$$

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