Use the table below to evaluate d/dx [f[g(2x)]] at x = 1

x 1 2 3 4
f(x) 6 1 8 2
f '(x) 6 1 8 2
g(x) 1 4 4 3
g '(x) 9 5 5 -4

To evaluate d/dx [f[g(2x)]] at x = 1, we need to follow several steps:

Step 1: Substitute x = 1 into the expression f[g(2x)]:
f[g(2x)] = f[g(2*1)] = f[g(2)] = f[g(2)].

Step 2: Substitute x = 1 into the g(x) table to find g(2):
g(x) -> g(2)
1 -> ?
4 -> ?
4 -> ?
3 -> ?

Looking at the g(x) table, we can see that when x = 2, g(x) = 4. Therefore, g(2) = 4.

Step 3: Substitute g(2) = 4 into f(x) table to find f[g(2)]:
f(x) -> f[g(2)]
6 -> ?
1 -> ?
8 -> ?
2 -> ?

Looking at the f(x) table, we can see that when x = 4, f(x) = 2. Therefore, f[g(2)] = f(4) = 2.

Step 4: Calculate d/dx [f[g(2x)]] by taking the derivative with respect to x of f[g(2x)]:
d/dx [f[g(2x)]] = d/dx [f(4)].

Step 5: Finally, substitute x = 1 into the f '(x) table to find f '(4):
f '(x) -> f '(4)
6 -> ?
1 -> ?
8 -> ?
2 -> ?

Looking at the f '(x) table, we can see that when x = 4, f '(x) = 2. Therefore, f '(4) = 2.

Thus, d/dx [f[g(2x)]] at x = 1 is equal to f '(4), which is 2.