Question

How to derivative f(x)?

Let f and g be the functions in the table below. (a) If F(x) = f(f(x)), find F'(1). F'(1) = ? (b) If G(x) = g(g(x)), find G*(3).

Answer 1

To find the derivatives of compositions of functions, we use the chain rule. For F'(1), substitute x=1 into the chain rule equation to find the derivative. For G*(3), substitute x=3 into the chain rule equation to find the derivative.

Step-by-step explanation:

To find the derivative of a function, we need to use the rules of calculus. For part (a), to find F'(1), we need to find the derivative of F(x) = f(f(x)). Let's say f(x) = u and f(f(x)) = v. To find F'(x), we can use the chain rule: F'(x) = f'(f(x)) * f'(x). We substitute x=1 into the equation to find F'(1).

For part (b), to find G*(3), we need to find the derivative of G(x) = g(g(x)). Similarly, let's say g(x) = u and g(g(x)) = v. Again, we use the chain rule: G'(x) = g'(g(x)) * g'(x). We substitute x=3 into the equation to find G*(3).