3t 2t - 1 1-t 1+t [][] 3 and B(t) = 51² 51³ t Verify the product law for differentiation, (AB)' = A'B + AB' where A(t) = To calculate (AB)', first calculate AB. AB=
3t 2t - 1 1-t 1+t [][] 3 and B(t) = 51² 51³ t Verify the product law for differentiation, (AB)' = A'B + AB' where A(t) = To calculate (AB)', first calculate AB. AB=
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Verification of the Product Law for Differentiation**
To verify the product law for differentiation, \((AB)' = A'B + AB'\), we need to work with the following matrices \(A(t)\) and \(B(t)\):
\[ A(t) = \begin{bmatrix}
3t & 2t - 1 \\
t & \frac{3}{t}
\end{bmatrix} \]
\[ B(t) = \begin{bmatrix}
1 - t & 1 + t \\
5t^2 & 5t^3
\end{bmatrix} \]
---
**Step 1: Calculate \(AB\)**
To calculate \((AB)'\), we first need to calculate \(AB\).
\[ AB = \begin{bmatrix}
3t & 2t - 1 \\
t & \frac{3}{t}
\end{bmatrix}
\begin{bmatrix}
1 - t & 1 + t \\
5t^2 & 5t^3
\end{bmatrix} \]
This involves multiplying the two matrices:
\[ AB = \begin{bmatrix}
(3t)(1-t) + (2t-1)(5t^2) & (3t)(1+t) + (2t-1)(5t^3) \\
(t)(1-t) + \left(\frac{3}{t}\right)(5t^2) & (t)(1+t) + \left(\frac{3}{t}\right)(5t^3)
\end{bmatrix} \]
Perform the matrix multiplication step-by-step:
\[
AB = \begin{bmatrix}
3t - 3t^2 + 10t^3 - 5t^2 & 3t + 3t^2 - 5t^3 + 10t^3 - 5t^3 \\
t - t^2 + 15t & t + t^2 + 15t^2
\end{bmatrix}
\]
Combine like terms:
\[ AB = \begin{bmatrix}
10t^3 - 8t^2 + 3t & 8t^3 + 3t + 10t^3 \\
16t - t^2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06bbd155-c9d1-4bdc-8881-0cd893ac016a%2F2b2967ee-1225-4e6e-9740-26593e806b08%2F1qv0aoc_processed.png&w=3840&q=75)
Transcribed Image Text:---
**Verification of the Product Law for Differentiation**
To verify the product law for differentiation, \((AB)' = A'B + AB'\), we need to work with the following matrices \(A(t)\) and \(B(t)\):
\[ A(t) = \begin{bmatrix}
3t & 2t - 1 \\
t & \frac{3}{t}
\end{bmatrix} \]
\[ B(t) = \begin{bmatrix}
1 - t & 1 + t \\
5t^2 & 5t^3
\end{bmatrix} \]
---
**Step 1: Calculate \(AB\)**
To calculate \((AB)'\), we first need to calculate \(AB\).
\[ AB = \begin{bmatrix}
3t & 2t - 1 \\
t & \frac{3}{t}
\end{bmatrix}
\begin{bmatrix}
1 - t & 1 + t \\
5t^2 & 5t^3
\end{bmatrix} \]
This involves multiplying the two matrices:
\[ AB = \begin{bmatrix}
(3t)(1-t) + (2t-1)(5t^2) & (3t)(1+t) + (2t-1)(5t^3) \\
(t)(1-t) + \left(\frac{3}{t}\right)(5t^2) & (t)(1+t) + \left(\frac{3}{t}\right)(5t^3)
\end{bmatrix} \]
Perform the matrix multiplication step-by-step:
\[
AB = \begin{bmatrix}
3t - 3t^2 + 10t^3 - 5t^2 & 3t + 3t^2 - 5t^3 + 10t^3 - 5t^3 \\
t - t^2 + 15t & t + t^2 + 15t^2
\end{bmatrix}
\]
Combine like terms:
\[ AB = \begin{bmatrix}
10t^3 - 8t^2 + 3t & 8t^3 + 3t + 10t^3 \\
16t - t^2
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