Answer: The determinant of the given matrix is -75e^24x.
Explanation:
To evaluate the determinant of a matrix containing functions, we can use a method called Laplace Expansion. This method involves taking the determinant of the matrix by expanding it along its first row.
Using Laplace Expansion, the determinant of the given matrix can be calculated as follows:
|e^5x 3^2x 3e^6x 5e^3x| = e^5x(3^2x5e^3x - 3e^6x3^2x) = e^5x(45e^9x - 27e^12x) = e^5x(18e^12x) = 18e^17x = -75e^24x.
Therefore, the determinant of the given matrix is -75e^24x.