The graphs of f and g are shown. Let
h(x) = f(g(x))
and
s(x) = g(f(x)).
f(-1)= 4
f'(x)=1
g(-1)=5
g'(-1)=-1/2
f(8)= 6
f'(x)= 2
g(x)= 2
g'(x)=-1/2
(a) Find
h'(−1).
(b) Find
s'(8).
To find h'(−1), we need to use the chain rule, which states that if a function is composed of two other functions, then the derivative of the composition is the derivative of the outer function times the derivative of the inner function.
(a) Finding h'(−1):
Step 1: Identify the inner function and its derivative.
The inner function in h(x) = f(g(x)) is g(x), and its derivative is g'(-1) = -1/2.
Step 2: Identify the outer function and its derivative.
The outer function in h(x) = f(g(x)) is f(x), and its derivative is f'(x) = 1 (given).
Step 3: Apply the chain rule.
The chain rule states that h'(x) = f'(g(x)) * g'(x). Since we need to find h'(-1), we substitute -1 for x:
h'(-1) = f'(g(-1)) * g'(-1).
Step 4: Evaluate the derivatives at the given values.
Substituting the given values, we get:
h'(-1) = f'(g(-1)) * g'(-1) = f'(5) * (-1/2).
However, we don't have the value of f'(5), so the answer cannot be determined with the given information.
(b) Finding s'(8):
Step 1: Identify the inner function and its derivative.
The inner function in s(x) = g(f(x)) is f(x), and its derivative is f'(x) = 2.
Step 2: Identify the outer function and its derivative.
The outer function in s(x) = g(f(x)) is g(x), and its derivative is g'(x) = -1/2 (given).
Step 3: Apply the chain rule.
The chain rule states that s'(x) = g'(f(x)) * f'(x). Since we need to find s'(8), we substitute 8 for x:
s'(8) = g'(f(8)) * f'(8).
Step 4: Evaluate the derivatives at the given values.
Substituting the given values, we get:
s'(8) = g'(f(8)) * f'(8) = g'(6) * 2.
However, we don't have the value of g'(6), so the answer cannot be determined with the given information.