Differentiating and integrating determinants is one of the integral concepts in mathematics. This lesson will cover the steps on how to differentiate and integrate determinants easily using several solved example questions.

All Contents in Determinants

## Differentiation of Determinants

Let

$$\begin{array}{l}\Delta \left( x \right)=\left| \begin{matrix} {{f}_{1}}\left( x \right) & {{g}_{1}}\left( x \right) \\ {{f}_{2}}\left( x \right) & {{g}_{2}}\left( x \right) \\ \end{matrix} \right|,\;\;where \;\;{{f}_{1}}\left( x \right),{{f}_{2}}\left( x \right),{{g}_{1}}\left( x \right)\;\; and \;\;{{g}_{2}}\left( x \right)\end{array}$$

are functions of x. Then,

$$\begin{array}{l}\Delta ‘\left( x \right)=\left| \begin{matrix} {{f}_{1}}’\left( x \right) & {{g}_{1}}’\left( x \right) \\ {{f}_{2}}\left( x \right) & {{g}_{2}}\left( x \right) \\ \end{matrix} \right|+\left| \begin{matrix} {{f}_{1}}\left( x \right) & {{g}_{1}}\left( x \right) \\ {{f}_{2}}’\left( x \right) & {{g}_{2}}’\left( x \right) \\ \end{matrix} \right| \;\;\;Also, \;\;\;\Delta ‘\left( x \right)=\left| \begin{matrix} {{f}_{1}}’\left( x \right) & {{g}_{1}}\left( x \right) \\ {{f}_{2}}’\left( x \right) & {{g}_{2}}\left( x \right) \\ \end{matrix} \right|+\left| \begin{matrix} {{f}_{1}}\left( x \right) & {{g}_{1}}’\left( x \right) \\ {{f}_{2}}\left( x \right) & {{g}_{2}}’\left( x \right) \\ \end{matrix} \right|\end{array}$$

### How to Differentiate a Determinant?

Thus, to differentiate a determinant, we differentiate one row (or column) at a time, keeping others unchanged. If we write

$$\begin{array}{l}\Delta \left( x \right)=\left[ {{C}_{1}}\,\,\,{{C}_{2}} \right],\end{array}$$
where Ci denotes the ith column, then
$$\begin{array}{l}\Delta ‘\left( x \right)=\left[ {{C}_{1}}’\,\,\,{{C}_{2}} \right]+\left[ {{C}_{1}}\,\,\,{{C}_{2}}’ \right],\end{array}$$
where Ci ‘ denotes the column obtained by differentiating functions in the ith column Ci. Also, if
$$\begin{array}{l}\Delta \left( x \right)=\left[ \begin{matrix} {{R}_{1}} \\ {{R}_{2}} \\ \end{matrix} \right],\; then \;\Delta ‘\left( x \right)=\left[ \begin{matrix} {{R}_{1}}’ \\ {{R}_{2}} \\ \end{matrix} \right]+\left[ \begin{matrix} {{R}_{1}} \\ {{R}_{2}}’ \\ \end{matrix} \right]\end{array}$$

Similarly, we can differentiate determinants of higher order.

Note: Differentiation can also be done column-wise by taking one column at a time.

## Integration of Determinants

If f(x), g(x) and h(x) are functions of x and a, b, c, α, β and γ are constants such that

$$\begin{array}{l}\Delta \left( x \right)=\left| \begin{matrix} f\left( x \right) & g\left( x \right) & h\left( x \right) \\ a & b & c \\ \alpha & \beta & \gamma \\ \end{matrix} \right|,\end{array}$$

then the integral of the determinants is given by i.e.

$$\begin{array}{l}\int{\Delta \left( x \right)dx=\left| \begin{matrix} \int{f\left( x \right)dx} & \int{g\left( x \right)dx} & \int{h\left( x \right)dx} \\ a & b & c \\ \alpha & \beta & \gamma \\ \end{matrix} \right|}\end{array}$$

### Example Problems on Differentiation and Integration of Determinants

Example 1: If

$$\begin{array}{l}\Delta \left( x \right)=\left| \begin{matrix} {{\sin }^{2}}x & \log \cos x & \log \tan x \\ {{n}^{2}} & 2n-1 & 2n+1 \\ 1 & -2\log 2 & 0 \\ \end{matrix} \right|,\; then \;evaluate\; \int\limits_{0}^{\pi /2}{\Delta \left( x \right)dx.}\end{array}$$

Solution:

By applying integration on variable elements of determinant we will solve the given problem.

We have

$$\begin{array}{l}\Delta \left( x \right)=\left| \begin{matrix} {{\sin }^{2}}x & \log \cos x & \log \tan x \\ {{n}^{2}} & 2n-1 & 2n+1 \\ 1 & -2\log 2 & 0 \\ \end{matrix} \right|;\,\,\int\limits_{0}^{\pi /2}{\Delta \left( x \right)dx=\left| \begin{matrix} \int\limits_{0}^{\pi /2}{{{\sin }^{2}}x\,dx} & \int\limits_{0}^{\pi /2}{\log \,\cos x\,dx} & \int\limits_{0}^{\pi /2}{\log \tan x\,dx} \\ {{n}^{2}} & 2n-1 & 2n+1 \\ 1 & -2\log 2 & 0 \\ \end{matrix} \right|}\end{array}$$

$$\begin{array}{l}=\left| \begin{matrix} \frac{\pi }{4} & -\frac{\pi }{2}\log 2 & 0 \\ {{n}^{2}} & 2n-1 & 2n+1 \\ 1 & -2\log 2 & 0 \\ \end{matrix} \right|\end{array}$$

= (π/2) 2n log 2 +  (π/2) log 2 –  (π/2) 2n log 2 –  (π/2) log 2

= 0

Example 2: If

$$\begin{array}{l}f\left( x \right)=\left| \begin{matrix} {{x}^{n}} & \sin x & \cos x \\ n! & \sin \frac{n\pi }{2} & \cos \frac{n\pi }{2} \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix} \right|,\; then \;show \;that \;\frac{{{d}^{n}}}{d{{x}^{n}}}\left\{ f\left( x \right) \right\}=0 \;at \;\;x=0.\end{array}$$

Solution:

By applying differentiation on variable elements of the determinant we will solve the given problem.

We have,

$$\begin{array}{l}f\left( x \right)=\left| \begin{matrix} {{x}^{n}} & \sin x & \cos x \\ n! & \sin \frac{n\pi }{2} & \cos \frac{n\pi }{2} \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix} \right|; \;\;\frac{{{d}^{n}}}{d{{x}^{n}}}\left\{ f\left( x \right) \right\}=\left| \begin{matrix} \frac{{{d}^{n}}}{d{{x}^{n}}}\left( {{x}^{n}} \right) & \frac{{{d}^{n}}}{d{{x}^{n}}}\left( \sin x \right) & \frac{{{d}^{n}}}{d{{x}^{n}}}\left( \cos x \right) \\ n! & \sin \frac{n\pi }{2} & \cos \frac{n\pi }{2} \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix} \right|\end{array}$$

$$\begin{array}{l}=\left| \begin{matrix} n! & \sin \left( x+\frac{n\pi }{2} \right) & \cos \left( x+\frac{n\pi }{2} \right) \\ n! & \sin \frac{n\pi }{2} & \cos \frac{n\pi }{2} \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix} \right|; \;\;\;\;{{\left( \frac{{{d}^{n}}}{d{{x}^{n}}}\left\{ f\left( x \right) \right\} \right)}_{x=0}}=\left| \begin{matrix} n! & \sin \frac{n\pi }{2} & \cos \frac{n\pi }{2} \\ n! & \sin \frac{n\pi }{2} & \cos \frac{n\pi }{2} \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix} \right|=0\end{array}$$

### Problem Solving Tactics

Let

$$\begin{array}{l}\Delta \left( x \right)=\left| \begin{matrix} {{f}_{1}}\left( x \right) & {{f}_{2}}\left( x \right) & {{f}_{3}}\left( x \right) \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\ \end{matrix} \right|,\;\;\; then \;\;\; \Delta ‘\left( x \right)=\left| \begin{matrix} {{f}_{1}}’\left( x \right) & {{f}_{2}}’\left( x \right) & {{f}_{3}}’\left( x \right) \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\ \end{matrix} \right|,\end{array}$$

and in general

$$\begin{array}{l}{{\Delta }^{n}}\left( x \right)=\left| \begin{matrix} f_{1}^{n}\left( x \right) & f_{2}^{n}\left( x \right) & f_{3}^{n}\left( x \right) \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\ \end{matrix} \right|\end{array}$$
where n is any positive integer and
$$\begin{array}{l}{{f}^{n}}\left( x \right)\end{array}$$
denotes the nth derivative of
$$\begin{array}{l}f\left( x \right).\end{array}$$

Let

$$\begin{array}{l}\Delta \left( x \right)=\left| \begin{matrix} f\left( x \right) & g\left( x \right) & h\left( x \right) \\ a & b & c \\ l & m & n \\ \end{matrix} \right|,\end{array}$$

where a,b,c,l,m and n are constants.

$$\begin{array}{l}\Rightarrow \,\,\int\limits_{a}^{b}{\Delta \left( x \right)dx=\left| \begin{matrix} \int\limits_{a}^{b}{f\left( x \right)dx} & \int\limits_{a}^{b}{g\left( x \right)dx} & \int\limits_{a}^{b}{h\left( x \right)dx} \\ a & b & c \\ l & m & n \\ \end{matrix} \right|}\end{array}$$

If the elements of more than one column or rows are functions of x then the integration can be done only after evaluation/expansion of the determinant.