Let f and g be the functions in the table below.
x f(x) g(x) f '(x) g'(x)
1 3 2 4 6
2 1 3 5 7
3 2 1 7 9
(a) If F(x) = f(f(x)), find F '(2).
(b) If G(x) = g(g(x)), find G '(1).
I know I probably have to use the chain rule. But how?
I figured it out. Thanks, though.
To solve both parts (a) and (b), we'll indeed use the chain rule, which states that the derivative of a composition of functions is the product of the derivative of the outer function and the derivative of the inner function.
(a) Let's find F '(x) first by applying the chain rule. We have F(x) = f(f(x)).
The inner function is f(x), and its derivative is f '(x). The outer function is f(u), where u = f(x). The derivative of the outer function with respect to u is f'(u). So, we have:
F '(x) = f'(f(x)) * f '(x)
To find F '(2), we substitute x = 2:
F '(2) = f'(f(2)) * f '(2)
To find f(2), look at the table given and find the value of f(x) when x = 2. From the table, we see that f(2) = 1.
Similarly, to find f'(2), look at the table and find the value of f'(x) when x = 2. We see that f'(2) = 5.
So, we substitute these values into our expression for F '(2):
F '(2) = f'(f(2)) * f '(2) = f'(1) * 5
From the table, we see that f(1) = 3. Therefore, f '(1) = 4.
Substituting these values into the expression, we have:
F '(2) = f'(1) * 5 = 4 * 5 = 20
(b) Similarly, let's find G '(x) by applying the chain rule. We have G(x) = g(g(x)).
The inner function is g(x), and its derivative is g '(x). The outer function is g(u), where u = g(x). The derivative of the outer function with respect to u is g'(u). So, we have:
G '(x) = g'(g(x)) * g '(x)
To find G '(1), we substitute x = 1:
G '(1) = g'(g(1)) * g '(1)
To find g(1), look at the table and find the value of g(x) when x = 1. From the table, we see that g(1) = 2.
Similarly, to find g'(1), look at the table and find the value of g'(x) when x = 1. We see that g'(1) = 6.
So, we substitute these values into our expression for G '(1):
G '(1) = g'(g(1)) * g '(1) = g'(2) * 6
From the table, we see that g(2) = 3. Therefore, g '(2) = 7.
Substituting these values into the expression, we have:
G '(1) = g'(2) * 6 = 7 * 6 = 42
To summarize:
(a) F '(2) = 20
(b) G '(1) = 42
To find the derivatives of composite functions using the chain rule, follow these steps:
(a) If F(x) = f(f(x)), find F '(2).
Step 1: Recall the chain rule, which states that if F(x) = f(g(x)), then F '(x) = f '(g(x)) * g '(x).
Step 2: Apply the chain rule to find the derivative of F(x) = f(f(x)):
F '(x) = f '(f(x)) * f '(x).
Step 3: Substitute x = 2 into the derivative expression:
F '(2) = f '(f(2)) * f '(2).
Step 4: Use the table to find the values of f(x) and f '(x) at x = 2:
f(2) = 1 and f '(2) = 5.
Step 5: Substitute the values into the expression:
F '(2) = f '(f(2)) * f '(2) = f '(1) * 5.
Step 6: Use the table to find the value of f '(x) at x = 1:
f '(1) = 4.
Step 7: Substitute the value into the expression:
F '(2) = f '(f(2)) * f '(2) = f '(1) * 5 = 4 * 5 = 20.
(b) If G(x) = g(g(x)), find G '(1).
Step 1: Apply the chain rule to find the derivative of G(x) = g(g(x)):
G '(x) = g '(g(x)) * g '(x).
Step 2: Substitute x = 1 into the derivative expression:
G '(1) = g '(g(1)) * g '(1).
Step 3: Use the table to find the values of g(x) and g '(x) at x = 1:
g(1) = 3 and g '(1) = 6.
Step 4: Substitute the values into the expression:
G '(1) = g '(g(1)) * g '(1) = g '(3) * 6.
Step 5: Use the table to find the value of g '(x) at x = 3:
g '(3) = 9.
Step 6: Substitute the value into the expression:
G '(1) = g '(g(1)) * g '(1) = g '(3) * 6 = 9 * 6 = 54.
Therefore, F '(2) = 20 and G '(1) = 54.