Differentiation Of a Determinant

Let $ \large \Delta (x) = \left| \begin{array}{cc} a_1(x) & b_1(x) \\ a_2(x) & b_2(x) \end{array} \right| $

Then , $ \large \Delta’ (x) = \left| \begin{array}{cc} a_1′(x) & b_1′(x) \\ a_2(x) & b_2(x) \end{array} \right| + \left| \begin{array}{cc} a_1(x) & b_1(x) \\ a_2′(x) & b_2′(x) \end{array} \right|$

Where a dash (‘) denotes derivative with respect to x.

If we write $ \large \Delta (x) = \left| \begin{array}{ccc} C_1 C_2 C_3 \end{array} \right| $

Then , $ \large \Delta’ (x) = \left| \begin{array}{ccc} C_1′ C_2 C_3 \end{array} \right| + \left| \begin{array}{ccc} C_1 C_2′ C_3 \end{array} \right| + \left| \begin{array}{ccc} C_1 C_2 C_3′ \end{array} \right| $

Similarly if , $ \large \Delta (x) = \left| \begin{array}{c} R_1 \\ R_2 \\ R_3 \end{array} \right| $

Then $ \large \Delta’ (x) = \left| \begin{array}{c} R_1′ \\ R_2 \\ R_3 \end{array} \right| + \left| \begin{array}{c} R_1 \\ R_2′ \\ R_3 \end{array} \right| + \left| \begin{array}{c} R_1 \\ R_2 \\ R_3′ \end{array} \right|$

Illustration : If f , g , h are differentiable functions of x , and

$ \large \Delta (x) = \left| \begin{array}{ccc} f & g & h \\ (x f)’ & (x g)’ & (x h)’ \\ (x^2 f)” & (x^2 g)” & (x^2 h)” \end{array} \right| $

Prove that :  $ \large \Delta ‘(x) = \left| \begin{array}{ccc} f & g & h \\ f’ & g’ & h’ \\ (x^3 f”)’ & (x^3 g”)’ & (x^3 h”)’ \end{array} \right| $

Solution: $ \large \Delta (x) = \left| \begin{array}{ccc} f & g & h \\ x f’ & x g’ + g & x h’ + h \\ 4 x f’ + 2 f + x^2 f” & 4 x g’ + 2 g + x^2 g” & 4 x h’ + 2 h + x^2 h” \end{array} \right| $

Operating (i) R2 → R2 − R1 (ii) R3 → R3 − 4 R2 + 2R1 and shifting x of R2 to R3

$ \large \Delta (x) = \left| \begin{array}{ccc} f & g & h \\ f’ & g’ & h’ \\ x^3 f” & x^3 g” & x^3 h” \end{array} \right| $

$ \large \Delta ‘(x) = 0 + 0 + \left| \begin{array}{ccc} f & g & h \\ f’ & g’ & h’ \\ (x^3 f”)’ & (x^3 g”)’ & (x^3 h”)’ \end{array} \right| $

Hence Proved .

Exercise :

(i) If fr (x), gr (x), hr (x) where r = 1, 2, 3 are polynomials in x such that fr(a)= gr (a) = hr(a) , r = 1 , 2 , 3 and $ \large F (x) = \left| \begin{array}{ccc} 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{array} \right| $ , then find F’ (a).

(ii) Let $ \large f (x) = \left| \begin{array}{ccc} cosx & x & 1 \\ 2sinx & x^2 & 2x \\ tanx & x & 1 \end{array} \right| $ . Find $\large \lim_{x\rightarrow 0} \frac{f'(x)}{x}$

(iii) If $ \large \Delta = \left| \begin{array}{ccc} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x \end{array} \right| $ , then find $\large \frac{d\Delta}{dx}$

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