Verify the product law for differentiation, (AB)' =A'B +AB' where A(t)= To calculate (AB)', first calculate AB. AB= www 3t 3t-1 3 and B(t) = 1-t 1+t 2t² 5t³
Verify the product law for differentiation, (AB)' =A'B +AB' where A(t)= To calculate (AB)', first calculate AB. AB= www 3t 3t-1 3 and B(t) = 1-t 1+t 2t² 5t³
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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### Verification of Product Law for Differentiation
To verify the product law for differentiation, we need to show that:
\[ (AB)' = A'B + AB' \]
where the matrices \( A(t) \) and \( B(t) \) are given by:
\[
A(t) =
\begin{bmatrix}
3t & 3t - 1 \\
t & \frac{3}{t}
\end{bmatrix}
\]
and
\[
B(t) =
\begin{bmatrix}
1 - t & 1 + t \\
2t^2 & 5t^3
\end{bmatrix}
\]
---
### Steps to Calculate \( (AB)' \):
**1. Calculate \( AB \)**
To calculate \( (AB)' \), first calculate the product \( AB \):
\[ AB = \begin{bmatrix} \text{[blank space]} \end{bmatrix} \]
(Note: The exact calculation steps for \( AB \) need to be provided based on matrix multiplication rules.)
---
Let's proceed with these steps to verify the product law for differentiation in the context of matrix calculus.
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Transcribed Image Text:---
### Verification of Product Law for Differentiation
To verify the product law for differentiation, we need to show that:
\[ (AB)' = A'B + AB' \]
where the matrices \( A(t) \) and \( B(t) \) are given by:
\[
A(t) =
\begin{bmatrix}
3t & 3t - 1 \\
t & \frac{3}{t}
\end{bmatrix}
\]
and
\[
B(t) =
\begin{bmatrix}
1 - t & 1 + t \\
2t^2 & 5t^3
\end{bmatrix}
\]
---
### Steps to Calculate \( (AB)' \):
**1. Calculate \( AB \)**
To calculate \( (AB)' \), first calculate the product \( AB \):
\[ AB = \begin{bmatrix} \text{[blank space]} \end{bmatrix} \]
(Note: The exact calculation steps for \( AB \) need to be provided based on matrix multiplication rules.)
---
Let's proceed with these steps to verify the product law for differentiation in the context of matrix calculus.
---
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