Question

If $f\left(x\right)=\left|\begin{array}{ccc}2x& x^2& x^3\\ x^2+2x& 1& 3x+1\\ 2x& 1-3x^2& 5x\end{array}\right|$, then find $f^{\prime }\left(1\right)$

Answer 1

Que. $f\left(x\right)=\left|\begin{array}{ccc}2x& x^2& x^3\\ x^2+2x& 1& 3x+1\\ 2x& 1-3x^2& 5x\end{array}\right|$, then $f^{\prime }\left(1\right)$

$f\left(x\right)=2x(5x-(3x+1\left)\right(1-3x^2\left)\right)-x^2\left(5x\right(x^2+2x)-2x(3x+1\left)\right)+x^3\left(\right(x^2+2x\left)\right(1-3x^2)-2x)$

$=2x(5x-(3x-9x^3+1-3x^2\left)\right)-x^2(5x^3+10x^2-6x^2-2x)+x^3(x^2-3x^4+2x-6x^3-2x)$

$=2x(2x+9x^3+3x^2-1)-x^2(5x^3+10x^2-6^2-2x)+x^3(-3x^4-6x^3+x^2)$

$=4x^2+18x^4+6x^3-2x-5x^5-10x^4+6x^4+2x^3-3x^7-6x^6+x^5$

$f\left(x\right)=-3x^7-6x^6-4x^5+14x^4+8x^3+4x^2-2x$

$f^1\left(x\right)=-3\times 7x^6-6\times 6x^5-4\times 5x^4+14\times 4x^3+8\times 3x^2+4\times 2x-2$

$f^1\left(x\right)=-21\times x^6-36x^5-20x^4+56x^3+24x^2+8x-2$

$f^1\left(1\right)=-21-36-20+56+24+8-2$

$=-79+88$

$f^1\left(1\right)=9$