(5) The following talbe gives values for f,g,f', and gʻ: xf (x)| f' (x) g(x) g'(x) 4 3 2 1 -2 4 4 2 4 (a) If h (x) = f (9 (x)), find h' (2). (b) If H (x) = g (f (x)), find H' (2) (6) Suppose g is a twice differentiable , and f (x) = x²g (sin x). (a) Find f' (x) in terms of g, g'. (b) Find f" (x) in terms of g, g', and g". (7) Suppose x4 + yª = 5 (a) Use implicit differentiation to find y' = . Simplify your answer. d²y (b) Find y" da? . Simplify your answer.
(5) The following talbe gives values for f,g,f', and gʻ: xf (x)| f' (x) g(x) g'(x) 4 3 2 1 -2 4 4 2 4 (a) If h (x) = f (9 (x)), find h' (2). (b) If H (x) = g (f (x)), find H' (2) (6) Suppose g is a twice differentiable , and f (x) = x²g (sin x). (a) Find f' (x) in terms of g, g'. (b) Find f" (x) in terms of g, g', and g". (7) Suppose x4 + yª = 5 (a) Use implicit differentiation to find y' = . Simplify your answer. d²y (b) Find y" da? . Simplify your answer.
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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![The image contains mathematical problems related to calculus, specifically focusing on differentiation and the chain rule. Below is a transcription suitable for an educational website:
---
### Problem 5
The following table gives values for \( f, g, f', \) and \( g' \):
\[
\begin{array}{|c|c|c|c|c|}
\hline
x & f(x) & f'(x) & g(x) & g'(x) \\
\hline
0 & 4 & 3 & 2 & 1 \\
2 & 0 & -2 & 4 & 5 \\
4 & 2 & 4 & 0 & 2 \\
\hline
\end{array}
\]
(a) If \( h(x) = f(g(x)) \), find \( h'(2) \).
(b) If \( H(x) = g(f(x)) \), find \( H'(2) \).
---
### Problem 6
Suppose \( g \) is a twice-differentiable function, and \( f(x) = x^2 g(\sin x) \).
(a) Find \( f'(x) \) in terms of \( g \) and \( g' \).
(b) Find \( f''(x) \) in terms of \( g \), \( g' \), and \( g'' \).
---
### Problem 7
Suppose \( x^4 + y^4 = 5 \).
(a) Use implicit differentiation to find \( y' = \frac{dy}{dx} \). Simplify your answer.
(b) Find \( y'' = \frac{d^2y}{dx^2} \). Simplify your answer.
---
Each problem involves the application of differentiation techniques such as the chain rule, product rule, and implicit differentiation. The solutions require interpreting and manipulating derivative expressions within given functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1300845a-828a-484d-8f06-319293900125%2Fc1ae630a-e485-4831-8890-bae5c40eb7d9%2Fvgxhp3d_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains mathematical problems related to calculus, specifically focusing on differentiation and the chain rule. Below is a transcription suitable for an educational website:
---
### Problem 5
The following table gives values for \( f, g, f', \) and \( g' \):
\[
\begin{array}{|c|c|c|c|c|}
\hline
x & f(x) & f'(x) & g(x) & g'(x) \\
\hline
0 & 4 & 3 & 2 & 1 \\
2 & 0 & -2 & 4 & 5 \\
4 & 2 & 4 & 0 & 2 \\
\hline
\end{array}
\]
(a) If \( h(x) = f(g(x)) \), find \( h'(2) \).
(b) If \( H(x) = g(f(x)) \), find \( H'(2) \).
---
### Problem 6
Suppose \( g \) is a twice-differentiable function, and \( f(x) = x^2 g(\sin x) \).
(a) Find \( f'(x) \) in terms of \( g \) and \( g' \).
(b) Find \( f''(x) \) in terms of \( g \), \( g' \), and \( g'' \).
---
### Problem 7
Suppose \( x^4 + y^4 = 5 \).
(a) Use implicit differentiation to find \( y' = \frac{dy}{dx} \). Simplify your answer.
(b) Find \( y'' = \frac{d^2y}{dx^2} \). Simplify your answer.
---
Each problem involves the application of differentiation techniques such as the chain rule, product rule, and implicit differentiation. The solutions require interpreting and manipulating derivative expressions within given functions.
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