Question

Let f and g be the functions in the table below. x f(x) f '(x) g(x) g'(x) 1 3 4 2 6 2 1 5 3 7 3 2 7 1 9 (a) If f(x) = f(f(x)), find f '(1). f '(1) = (b) If g(x) = g(g(x)), find g'(3). g'(3) =

Answer 1

To find f'(1), differentiate the equation f(x) = f(f(x)) with respect to x. Substitute x = 1 to find f'(1) = 1.

Step-by-step explanation:

To find f'(1), we need to determine the derivative of f(x). Since f(x) = f(f(x)), this means that the value of f(x) after applying f(x) is equal to f(x) itself. Let's differentiate both sides of the equation with respect to x to find f'(x):

Let g(x) = f(x), then f(x) = g(g(x)).

Now, differentiate both sides of the equation:

g'(g(x)) * g'(x) = g'(x)

Since g(x) = f(x), we can rewrite the equation as:

f'(f(x)) * f'(x) = f'(x)

Simplifying, we get:

f'(f(x)) = 1

Now, substitute x = 1 into the equation:

f'(f(1)) = 1

Since f(1) = 3, we have:

f'(3) = 1