# Differentiation of Determinant

Here you will learn differentiation of determinant with example.

Let’s begin –

## 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.

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) = $$\begin{vmatrix} x + b^2 & bc \\ bc & x + c^2 \end{vmatrix}$$ + $$\begin{vmatrix} x + a^2 & ac \\ ac & x + c^2 \end{vmatrix}$$ + $$\begin{vmatrix} x + a^2 & ab \\ ab & x + b^2 \end{vmatrix}$$

$$\implies$$ f'(x) = {$$(x+b^2)(x+c^2) – b^2c^2$$} + {$$(x+a^2)(x+c^2) – a^2c^2$$} + {$$(x+a^2)(x+b^2) – a^2b^2$$}

$$\implies$$ 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$$a^2 + b^2 + c^2$$.