## To determine which function is increasing the fastest when x = 2, you can follow these steps:

1. Differentiate each function:

- For f(x) = x^2, the derivative f'(x) = 2x.

- For g(x) = e^(2x), the derivative g'(x) = 2e^(2x).

- For h(x) = ln(2x), the derivative h'(x) = 1/x.

2. Plug in x = 2 into the derivatives:

- f'(2) = 2(2) = 4

- g'(2) = 2e^(2*2) = 2e^4 ≈ 109.1

- h'(2) = 1/2 = 0.5

3. Compare the values obtained:

- Since g'(2) ≈ 109.1 is the largest value among f'(2) = 4 and h'(2) = 0.5, the function g(x) = e^(2x) is increasing the fastest at x = 2.

Therefore, g(x) = e^(2x) is the function increasing the fastest when x = 2.