Question

the following table lists the values of functions f and g, and of their derivatives, f and g, for the x-values 0 and 1.
x | f(x) | g(x) | f(x) | g(x)
0 | 7 | 3 | -3 | -4
1 | 6 | 0 | 1 | -2
evaluate \\(\frac{d}{dx}f\big(g(x)\big)\\) at \\(x = 1\\).

Explanation:

Step1: Recall Chain Rule

The chain rule states that \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\).

Step2: Evaluate at \(x = 1\)

First, find \(g(1)\) from the table: \(g(1)=0\).
Then, find \(f'(g(1)) = f'(0)\). From the table, \(f'(0)= - 3\).
Next, find \(g'(1)\) from the table: \(g'(1)=-2\).
Now, multiply \(f'(g(1))\) and \(g'(1)\): \((-3)\times(-2)=6\).

Answer:

\(6\)