#Differentiating and Integrating Determinants This lesson will provide an overview of the steps needed to differentiate and integrate determinants, with several example questions to help illustrate the process.

All Contents on Determinants

Introduction to Determinants

Minors and Cofactors

Properties of Determinants

System of Linear Equations using Determinants

Differentiation and Integration of Determinants

Standard Determinants

Differentiation of Determinants

Let $\Delta(x) = \left| \begin{matrix} f_1(x) & g_1(x) \ f_2(x) & g_2(x) \end{matrix} \right|$

If f1(x) = f2(x) and g1(x) = g2(x), then x is a solution to both equations.

(\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|)

What is the Process for Differentiating 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 $C_i$ denotes the $i^{th}$ column, then (\begin{array}{l}\Delta’\left( x \right)=\left[ {{C}_{1}}’,,,{{C}_{2}} \right]+\left[ {{C}_{1}},,,{{C}_{2}}’ \right],\end{array} ) where $C_i’$ denotes the column obtained by differentiating functions in the $i^{th}$ column $C_i$. 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 $a$, $b$, $c$, $\alpha$, $\beta$ and $\gamma$ are constants such that

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

The determinant integral is given by; (\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} )

Differentiation and Integration of Determinants: Example Problems

Example 1: Evaluate $\int_0^{\pi/2}\left| \begin{matrix} {\sin^2}x & \log \cos x & \log \tan x \ {{n}^{2}} & 2n-1 & 2n+1 \ 1 & -2\log 2 & 0 \ \end{matrix} \right|dx$.

Given:

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Solution:

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By applying integration to the variable elements of the determinant, we can solve the given problem.

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

(\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|)

-(π/2) 2n log 2 + (π/2) log 2

= 0

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

$$\frac{{{d}^{n}}}{d{{x}^{n}}}\left{ f\left( 0 \right) \right}=0$$

Given:

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Solution:

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By taking the derivative of each element of the determinant, we can 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 (\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|).

In general,

$$\left|\begin{matrix}{\Delta}^n(x) &=& \left|\begin{matrix}f_1^n(x) & f_2^n(x) & f_3^n(x)\ b_1 & b_2 & b_3\ c_1 & c_2 & c_3\end{matrix}\right|\end{matrix}\right|$$ where n is any positive integer and $f_n(x)$ denotes the nth derivative of $f(x)$.

Let (\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| )

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

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

If the elements of more than one column or row are functions of $x$, then the integration can only be done after evaluating/expanding the determinant.

Video Lessons

#Integration - Important Questions

Important Theorems of Differentiation for JEE

![Important Theorems of Differentiation for JEE]()

Frequently Asked Questions

What is the Process for Differentiating a Determinant?

To differentiate a determinant, we have to:

1. Differentiate one row or column at a time, keeping others unchanged
2. Add the determinants so obtained

Integrating a Determinant

If the elements of more than one column or row are functions of $x$, then we should evaluate/expand the determinant before integrating each element of the first row.

An application of integration is finding the area under a curve.

We use integration to find the area under the curve of a function that is integrated.