**Text Transcription for Educational Website**---Let \( a_{1,1}(x), a_{1,2}(x), \ldots \) be differentiable real/complex valued functions in \( x \). Let \( f(x) = \det(a_{i,i}) \). Find a formula for \( f'(x) \). Hence do #19 on Page 212.---**Explanation:**The text involves finding the derivative of a determinant of differentiable functions. It suggests that the reader apply this concept to a related task or problem #19, located on page 212 of the referenced material. This requires understanding differentiable functions, determinants, and basic differentiation techniques. ```plaintext4 Determinants19. If \( f(s) \) is given by the following determinants, without the expansion of \( f(x) \) find \(\frac{d}{dx} f(x)\)(a) \( f(x) = \begin{vmatrix} e^x & e^{-x} & 1 \\ e^x & -e^{-x} & 0 \\ e^x & -e^{-x} & x \end{vmatrix} \);(b) \( f(x) = \begin{vmatrix} \cos x & \sin x & \ln |x| \\ -\sin x & \cos x & \frac{1}{x} \\ -\cos x & -\sin x & -\frac{1}{x^2} \end{vmatrix} \).Ans. (a) \(-2\); (b) \(1/x + 2/x^3\).```Explanation of Determinants:- **Matrix (a):** A 3x3 matrix with elements that are exponential functions, including both positive and negative exponents, along with polynomial terms like \(x\).- **Matrix (b):** A 3x3 matrix that includes trigonometric functions (\(\cos x\) and \(\sin x\)), logarithmic functions (\(\ln |x|\)), and rational expressions such as \(1/x\) and \(-1/x^2\).The task involves finding the derivative of the determinant for both these matrices with respect to \(x\), but without explicitly expanding the determinant. The answers provided indicate the calculated derivatives for each of the given matrices.

here we did coloumn wise differentiations Here we do row wise differentiations.

19. If f(s) is given by the following determinants, without the expansion of f(x) find f(x) dæ e-x 1 COS x sin x In x (a) f(x) = -* 0 (b) f(x)= - sin x COs x -e e - COS x – sin x x2 Ans. (a) —2; (b) 1/х + 2/3.

19. If f(s) is given by the following determinants, without the expansion of f(x) find f(x) dæ e-x 1 COS x sin x In x (a) f(x) = -* 0 (b) f(x)= - sin x COs x -e e - COS x – sin x x2 Ans. (a) —2; (b) 1/х + 2/3.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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