Here you will learn differentiation of determinant with example.
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Differentiation of Determinant
To differentiate a determinant, we differerentiate one row (or column) at a time, keeping others unchanged.
for example, if
D(x) = \(\begin{vmatrix} f(x) & g(x) \\ u(x) & v(x) \end{vmatrix}\) , then
\(d\over dx\){D(x)} = \(\begin{vmatrix} f'(x) & g'(x) \\ u(x) & v(x) \end{vmatrix}\) + \(\begin{vmatrix} f(x) & g(x) \\ u'(x) & v'(x) \end{vmatrix}\)
Also,
\(d\over dx\){D(x)} = \(\begin{vmatrix} f'(x) & g(x) \\ u'(x) & v(x) \end{vmatrix}\) + \(\begin{vmatrix} f(x) & g'(x) \\ u(x) & v'(x) \end{vmatrix}\)
Similar results holds for the differentiation of determinant of higher order.ย
Example : If f(x) = \(\begin{vmatrix} x^2 + a^2 & a \\ x^2 + b^2 & b \end{vmatrix}\), find f'(x).
Solution : We have,
f(x) = \(\begin{vmatrix} x^2 + a^2 & a \\ x^2 + b^2 & b \end{vmatrix}\)
\(\implies\) f'(x) = \(\begin{vmatrix} 2x & 0 \\ x^2 + b^2 & b \end{vmatrix}\) + \(\begin{vmatrix} x^2 + a^2 & a \\ 2x & 0 \end{vmatrix}\)
\(\implies\) f'(x) = {2bx} + {2ax}
f'(x) = 2x(a +b)
Example : If f(x) = \(\begin{vmatrix} x + a^2 & ab & ac \\ ab & x + b^2 & bc \\ ac & bc & x + c^2 \end{vmatrix}\), find f'(x).
Solution : We have,
f(x) = \(\begin{vmatrix} x + a^2 & ab & ac \\ ab & x + b^2 & bc \\ ac & bc & x + c^2 \end{vmatrix}\)
\(\implies\) f'(x) = \(\begin{vmatrix} 1 & 0 & 0 \\ ab & x + b^2 & bc \\ ac & bc & x + c^2 \end{vmatrix}\) + \(\begin{vmatrix} x + a^2 & ab & ac \\ 0 & 1 & 0 \\ ac & bc & x + c^2 \end{vmatrix}\) + \(\begin{vmatrix} x + a^2 & ab & ac \\ ab & x + b^2 & bc \\ 0 & 0 & 1 \end{vmatrix}\)
\(\implies\) f'(x) = \(\begin{vmatrix}ย x + b^2 & bc \\ bc & x + c^2 \end{vmatrix}\) + \(\begin{vmatrix} x + a^2 & ac \\ ac & x + c^2 \end{vmatrix}\) + \(\begin{vmatrix} x + a^2 & ab \\ ab & x + b^2 \end{vmatrix}\)
\(\implies\) f'(x) = {\((x+b^2)(x+c^2) โ b^2c^2\)} + {\((x+a^2)(x+c^2) โ a^2c^2\)} + {\((x+a^2)(x+b^2) โ a^2b^2\)}
\(\implies\) f'(x) = \(x^2\) + x\(b^2+c^2\) + \(x^2\) + x\(a^2+c^2\) + \(x^2\) + x\(a^2+b^2\)
f'(x) = \(3x^2\) + 2x\(a^2 + b^2 + c^2\).
