1.
Write the composite function in the form
f(g(x)). [Identify the inner function
u = g(x) and the outer function y = f(u).]
y=
(g(x), f(u)) =
2.
Write the composite function in the form
f(g(x)). [Identify the inner function
u = g(x) and the outer function
y = f(u).]
y = 3squareoot 1 + 5x
(g(x), f(u)) = ?
Find the derivative dy/dx.
3.
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 '(3).
F '(3) =
(b) If G(x) = g(g(x)), find G'(2).
G'(2) =
3.
If h(x) = all square root over this4 + 3f(x) where
f(3) = 4 and f '(3) = 3, find h'(3).
h'(3) =?
1. To write the composite function in the form f(g(x)), we need to identify the inner function u = g(x) and the outer function y = f(u).
Example:
Given y = f(g(x)), we can rewrite it as (g(x), f(u)) = ?
2. Similarly, for this example y = 3√(1 + 5x), we need to identify the inner function u = g(x) and the outer function y = f(u).
Example:
Given y = f(g(x)), we can rewrite it as (g(x), f(u)) = ?
To find the derivative dy/dx, we need to differentiate the function y with respect to x using the chain rule.
3. Given the functions f and g in the table, we need to find the derivative of the composite functions F(x) = f(f(x)) and G(x) = g(g(x)).
(a) To find F'(3), we will first compute the derivative of f(x), then evaluate it at x = 3, and finally compute the derivative of F(x) using the chain rule.
Example:
F'(3) =
(b) Similarly, to find G'(2), we will follow the same steps as in part (a).
Example:
G'(2) =
4. For the function h(x) = √(4 + 3f(x)), where f(3) = 4 and f'(3) = 3, we need to find the derivative h'(3).
Example:
First, we need to find the derivative of h(x) using the chain rule. Then, we substitute x = 3 and evaluate h'(3).
h'(3) = ?