Question

Given f(1) = 1, f'(1) = 5, g(1) = 3, g'(1) = 4, f'(3) = 2, and g'(3) = 6, compute the following derivatives:

17. (d/dx)[f(g(x))] at x=1.

Answer 1

The derivative of the composition of the functions f after g, denoted as (d/dx)[f(g(x))] at x=1, is computed using the chain rule and is found to be 8.

Step-by-step explanation:

The question involves calculus and specifically asks us to find the derivative of the composition of two functions, f after g, at the point x = 1. To do this, we will employ the chain rule for differentiation. The formula for the derivative of a composition of functions (f(g(x))) is as follows: the derivative of f with respect to g (f'(g(x))) multiplied by the derivative of g with respect to x (g'(x)).

In this case, we want to find (d/dx)[f(g(x))] at x=1. Given that f'(1) = 5 and g'(1) = 4 from the question, we apply the chain rule:

(d/dx)[f(g(x))] at x=1 is (f'(g(1)) · g'(1) = f'(3) · 4, because g(1) = 3. Considering we are given f'(3) = 2 in the information, this simplifies this to 2 · 4 = 8.

So the derivative of f after g at x = 1 is 8.