Find the following using the table below. x 3 4 f(x) 3 5 1 f'(x) -5 2 -4 1 g(x) 1 4 2 3 g'(x) 3 2 -1 4 If m(x) = 1 2 25 If q(x) = f(x) g(x), then q'(3) . f(x) g(x)' NA -2 then m' (3) If p(x) = f(g(x)), then p'(3) = = = -9 7 4 OF 08
Find the following using the table below. x 3 4 f(x) 3 5 1 f'(x) -5 2 -4 1 g(x) 1 4 2 3 g'(x) 3 2 -1 4 If m(x) = 1 2 25 If q(x) = f(x) g(x), then q'(3) . f(x) g(x)' NA -2 then m' (3) If p(x) = f(g(x)), then p'(3) = = = -9 7 4 OF 08
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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![**Using Derivatives and Functions from a Table**
The task involves finding derivatives of functions using the provided table of values.
**Table:**
| \( x \) | 1 | 2 | 3 | 4 |
|---------|---|---|---|---|
| \( f(x) \) | 3 | 5 | 1 | -2 |
| \( f'(x) \) | -5 | 2 | -4 | 1 |
| \( g(x) \) | 1 | 4 | 2 | 3 |
| \( g'(x) \) | 3 | 2 | -1 | 4 |
**Problems and Solutions:**
1. **Function \( q(x) = f(x) \cdot g(x) \)**
Find \( q'(3) \).
Solution:
\[
q'(x) = f(x)g'(x) + g(x)f'(x)
\]
At \( x = 3 \):
\[
q'(3) = 1 \cdot (-1) + 2 \cdot (-4) = -9
\]
**Result: \( -9 \)**
2. **Function \( m(x) = \frac{f(x)}{g(x)} \)**
Find \( m'(3) \).
Solution:
\[
m'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}
\]
At \( x = 3 \):
\[
m'(3) = \frac{2 \cdot (-4) - 1 \cdot (-1)}{2^2} = \frac{-7}{4}
\]
**Result: \( -\frac{7}{4} \)**
3. **Function \( p(x) = f(g(x)) \)**
Find \( p'(3) \).
Solution:
Using the chain rule:
\[
p'(x) = f'(g(x)) \cdot g'(x)
\]
At \( x = 3 \):
\[
g(3) = 2, \quad f'(2) = 2, \quad g'(3) = -1
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffdc4bd04-a070-4f0a-8c81-81114a590c8c%2Ff6a24640-c43c-4c31-b8e9-c115fe6b1cbc%2Fwogtfs_processed.png&w=3840&q=75)
Transcribed Image Text:**Using Derivatives and Functions from a Table**
The task involves finding derivatives of functions using the provided table of values.
**Table:**
| \( x \) | 1 | 2 | 3 | 4 |
|---------|---|---|---|---|
| \( f(x) \) | 3 | 5 | 1 | -2 |
| \( f'(x) \) | -5 | 2 | -4 | 1 |
| \( g(x) \) | 1 | 4 | 2 | 3 |
| \( g'(x) \) | 3 | 2 | -1 | 4 |
**Problems and Solutions:**
1. **Function \( q(x) = f(x) \cdot g(x) \)**
Find \( q'(3) \).
Solution:
\[
q'(x) = f(x)g'(x) + g(x)f'(x)
\]
At \( x = 3 \):
\[
q'(3) = 1 \cdot (-1) + 2 \cdot (-4) = -9
\]
**Result: \( -9 \)**
2. **Function \( m(x) = \frac{f(x)}{g(x)} \)**
Find \( m'(3) \).
Solution:
\[
m'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}
\]
At \( x = 3 \):
\[
m'(3) = \frac{2 \cdot (-4) - 1 \cdot (-1)}{2^2} = \frac{-7}{4}
\]
**Result: \( -\frac{7}{4} \)**
3. **Function \( p(x) = f(g(x)) \)**
Find \( p'(3) \).
Solution:
Using the chain rule:
\[
p'(x) = f'(g(x)) \cdot g'(x)
\]
At \( x = 3 \):
\[
g(3) = 2, \quad f'(2) = 2, \quad g'(3) = -1
\
![Find the first derivative of \( z = \tan(\sin^2(t)) \).
\[
\frac{dz}{dt} = \, \boxed{}
\]
Note that the system does not like \(\sin^2 x\) notation. Instead, you must input as \(\sin^2(x)\) if needed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffdc4bd04-a070-4f0a-8c81-81114a590c8c%2Ff6a24640-c43c-4c31-b8e9-c115fe6b1cbc%2F5glnqk6_processed.png&w=3840&q=75)
Transcribed Image Text:Find the first derivative of \( z = \tan(\sin^2(t)) \).
\[
\frac{dz}{dt} = \, \boxed{}
\]
Note that the system does not like \(\sin^2 x\) notation. Instead, you must input as \(\sin^2(x)\) if needed.
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