To solve the given problems, we need to utilize the chain rule in calculus for finding the derivatives of composite functions.
### Part (a)
Given the function [tex]\( h(x) = f(g(x)) \)[/tex], the derivative [tex]\( h'(x) \)[/tex] can be found using the chain rule:
[tex]\[ h'(x) = f'(g(x)) \cdot g'(x) \][/tex]
To find [tex]\( h'(1) \)[/tex]:
1. Determine [tex]\( g(1) \)[/tex].
From the table, [tex]\( g(1) = 2 \)[/tex].
2. Now, find [tex]\( f'(g(1)) \)[/tex].
Since [tex]\( g(1) = 2 \)[/tex], we need [tex]\( f'(2) \)[/tex] from the table. According to the table:
[tex]\( f'(2) = 5 \)[/tex].
3. Next, find [tex]\( g'(1) \)[/tex].
From the table, [tex]\( g'(1) = 6 \)[/tex].
4. Multiply these values together to get [tex]\( h'(1) \)[/tex]:
[tex]\[ h'(1) = f'(g(1)) \cdot g'(1) = 5 \cdot 6 = 30 \][/tex]
So, [tex]\( h'(1) = 30 \)[/tex].
### Part (b)
Given the function [tex]\( H(x) = g(f(x)) \)[/tex], the derivative [tex]\( H'(x) \)[/tex] can also be found using the chain rule:
[tex]\[ H'(x) = g'(f(x)) \cdot f'(x) \][/tex]
To find [tex]\( H'(1) \)[/tex]:
1. Determine [tex]\( f(1) \)[/tex].
From the table, [tex]\( f(1) = 3 \)[/tex].
2. Now, find [tex]\( g'(f(1)) \)[/tex].
Since [tex]\( f(1) = 3 \)[/tex], we need [tex]\( g'(3) \)[/tex] from the table. According to the table:
[tex]\( g'(3) = 9 \)[/tex].
3. Next, find [tex]\( f'(1) \)[/tex].
From the table, [tex]\( f'(1) = 4 \)[/tex].
4. Multiply these values together to get [tex]\( H'(1) \)[/tex]:
[tex]\[ H'(1) = g'(f(1)) \cdot f'(1) = 9 \cdot 4 = 36 \][/tex]
So, [tex]\( H'(1) = 36 \)[/tex].
In conclusion:
- [tex]\( h'(1) = 30 \)[/tex]
- [tex]\( H'(1) = 36 \)[/tex]
