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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
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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**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.
Transcribed Image Text:**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.
```plaintext 4 Determinants 19. 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.
Transcribed Image Text:```plaintext 4 Determinants 19. 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.
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