If fy-1-ya1b1 1-ya1b2 1-ya1b31-ya2b1 1-ya2b2 1-ya2b31-ya3b1 1-ya3b2 1-ya3b3 and a- b are even for i-1- 2- 3- then find f-2

If nbsp;f'(y)= a_1b_1(1 y)^a_1b_1 1 amp; a_2b_2(1 y)^a_1b_2 1 amp; a_3b_3(1 y)^a_1b_3 1 (1 y)^a_2b_1 amp;(1 y)^a_2b_2 amp;(1 y)^a_2b_3 (1 y)^a_3b_1 ...

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Byju's Answer

Standard XII

Mathematics

Derivative of a Determinant

If fy=1-ya1b1...

Question

If $f (y) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} (1 - y)^{a_{1} b_{1}} & (1 - y)^{a_{1} b_{2}} & (1 - y)^{a_{1} b_{3}} (1 - y)^{a_{2} b_{1}} & (1 - y)^{a_{2} b_{2}} & (1 - y)^{a_{2} b_{3}} (1 - y)^{a_{3} b_{1}} & (1 - y)^{a_{3} b_{2}} & (1 - y)^{a_{3} b_{3}} \end{matrix} ∣ ∣ ∣ ∣ ∣$ and $a, b$ are even for $i = 1, 2, 3$ . then find $f^{'} (2)$

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Solution

If $f^{'} (y) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} - a_{1} b_{1} (1 - y)^{a_{1} b_{1} - 1} & - a_{2} b_{2} (1 - y)^{a_{1} b_{2} - 1} & - a_{3} b_{3} (1 - y)^{a_{1} b_{3} - 1} (1 - y)^{a_{2} b_{1}} & (1 - y)^{a_{2} b_{2}} & (1 - y)^{a_{2} b_{3}} (1 - y)^{a_{3} b_{1}} & (1 - y)^{a_{3} b_{2}} & (1 - y)^{a_{3} b_{3}} \end{matrix} ∣ ∣ ∣ ∣ ∣$ + $∣ ∣ ∣ ∣ ∣ \begin{matrix} (1 - y)^{a_{1} b_{1}} & (1 - y)^{a_{1} b_{2}} & (1 - y)^{a_{1} b_{3}} - a_{2} b_{1} (1 - y)^{a_{2} b_{1} - 1} & - a_{2} b_{2} (1 - y)^{a_{2} b_{2} - 1} & - a_{2} b_{3} (1 - y)^{a_{2} b_{3} - 1} (1 - y)^{a_{3} b_{1}} & (1 - y)^{a_{3} b_{2}} & (1 - y)^{a_{3} b_{3}} \end{matrix} ∣ ∣ ∣ ∣ ∣$ + $∣ ∣ ∣ ∣ ∣ \begin{matrix} (1 - y)^{a_{1} b_{1}} & (1 - y)^{a_{1} b_{2}} & (1 - y)^{a_{1} b_{3}} (1 - y)^{a_{2} b_{1}} & (1 - y)^{a_{2} b_{2}} & (1 - y)^{a_{2} b_{3}} - a_{3} b_{1} (1 - y)^{a_{3} b_{1} - 1} & - a_{3} b_{2} (1 - y)^{a_{3} b_{2} - 1} & - a_{3} b_{3} (1 - y)^{a_{3} b_{3} - 1} \end{matrix} ∣ ∣ ∣ ∣ ∣$

$f^{'} (2) = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} b_{1} & a_{1} b_{2} & a_{1} b_{3} 1 & 1 & 1 1 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a_{2} b_{1} & a_{2} b_{2} & a_{2} b_{3} 1 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 1 & 1 & 1 a_{3} b_{1} & a_{3} b_{2} & a_{3} b_{3} \end{matrix} ∣ ∣ ∣ ∣$

$f^{'} (2) = 0 + 0 + 0 = 0$

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If f(y)=∣∣ ∣ ∣∣(1−y)a1b1(1−y)a1b2(1−y)a1b3(1−y)a2b1(1−y)a2b2(1−y)a2b3(1−y)a3b1(1−y)a3b2(1−y)a3b3∣∣ ∣ ∣∣ and a,b are even for i=1,2,3. then find f′(2)