Differentiation of Functions Expressed in Determinant Form

Neetesh Kumar | February 08, 2025 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Definition of Derivative of Functions Expressed in Determinant Form
• 2. Formula for Determinant Derivative
• 3. Solved Examples
• 4. Conclusion

Derivative of Functions Expressed in Determinant Form:

1. Let $f(x)$ = $\begin{vmatrix} f & g & h \\ u & v & w \\ l & m & n \end{vmatrix}$ where all functions are differentiable then

$F'(x) = \begin{vmatrix} f' & g' & h' \\ u & v & w \\ l & m & n \end{vmatrix} + \begin{vmatrix} f & g & h \\ u' & v' & w' \\ l & m & n \end{vmatrix} + \begin{vmatrix} f & g & h \\ u & v & w \\ l' & m' & n' \end{vmatrix}$

This result may be proved by the first principle and the same operation can also be done column wise.

2. When a function is expressed as a determinant, the derivative of such a function requires the application of specific properties of determinants and matrices.

If $f(x) = \det(A(x))$, where $A(x)$ is a matrix whose elements are functions of $x$, then the derivative of the determinant with respect to $x$ is given by:

$\boxed{\frac{d}{dx} \left[ \det(A(x)) \right] = \det(A(x)) \cdot \text{tr}\left(A^{-1}(x) \cdot A'(x)\right)}$

Where:

• $A(x)$ is the matrix.
• $A'(x)$ is the matrix obtained by differentiating each element of $A(x)$ with respect to $x$.
• $A^{-1}(x)$ is the inverse of the matrix $A(x)$.
• $\text{tr}(A)$ is the trace of matrix $A$, which is the sum of its diagonal elements.

Example 1: Derivative of a 2x2 Matrix Function

Let $A(x) = \begin{pmatrix} 2x & 3 \\ 4 & 5x \end{pmatrix}$. Find $\frac{d}{dx} \left[ \det(A(x)) \right]$.

Solution:

We first compute the determinant of $A(x)$:

$\det(A(x)) = (2x)(5x) - (4)(3) = 10x^2 - 12$.

Now, differentiate $\det(A(x))$:

$\frac{d}{dx} \left[ \det(A(x)) \right] = \frac{d}{dx} (10x^2 - 12) = 20x$.

Thus, the final answer is: $\boxed{20x}$.

Example 2: Derivative of a 3x3 Matrix Function

Let $A(x) = \begin{pmatrix} x^2 & 2x & 3 \\ 1 & x & 4 \\ 0 & 1 & x \end{pmatrix}$. Find $\frac{d}{dx} \left[ \det(A(x)) \right]$.

Solution:

First, compute the determinant of $A(x)$:

$\det(A(x)) = x^2 \begin{vmatrix} x & 4 \\ 1 & x \end{vmatrix} - 2x \begin{vmatrix} 1 & 4 \\ 0 & x \end{vmatrix} + 3 \begin{vmatrix} 1 & x \\ 0 & 1 \end{vmatrix}$.

Expanding the 2x2 determinants:

$\det(A(x)) = x^2 (x^2 - 4) - 2x (x) + 3 (1)$

$\det(A(x)) = x^4 - 4x^2 - 2x^2 + 3$

$\det(A(x)) = x^4 - 6x^2 + 3$.

Now, differentiate $\det(A(x))$:

$\frac{d}{dx} \left[ \det(A(x)) \right] = 4x^3 - 12x$

Thus, the final answer is: $\boxed{4x^3 - 12x}$.

Example 3: Derivative of a 2x2 Matrix Involving Exponential Function

Let $A(x) = \begin{pmatrix} e^x & x^2 \\ 3x & \ln(x) \end{pmatrix}$. Find $\dfrac{d}{dx} \left[ \det(A(x)) \right]$.

Solution:

First, compute the determinant of $A(x)$:

$\det(A(x)) = e^x \cdot \ln(x) - 3x \cdot x^2$

$\det(A(x)) = e^x \ln(x) - 3x^3$.

Now, differentiate $\det(A(x))$:

$\dfrac{d}{dx} \left[ \det(A(x)) \right] = \dfrac{d}{dx} \big( e^x. \ln(x) \big) - \dfrac{d}{dx} (3x^3)$

Using the product rule for $e^x \ln(x)$:

$\dfrac{d}{dx} \big( e^x \ln(x) \big) = e^x \ln(x) + e^x. \dfrac{1}{x}$

Differentiating $3x^3$:

$\frac{d}{dx} (3x^3) = 9x^2$

Thus, the derivative is:

$\frac{d}{dx} \left[ \det(A(x)) \right] = e^x \ln(x) + \frac{e^x}{x} - 9x^2$

Thus, the final answer is:

$\boxed{e^x \ln(x) + \frac{e^x}{x} - 9x^2}$.