Define a function f(x)=e^{x^2+6.9x} for all positive x. Denote f'(x) to be the derivative of f(x). What is the value of f'(0)?

To find the derivative of the function f(x) = e^{x^2 + 6.9x}, we will use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 1: Identify the outer and inner functions. Here, the outer function is e^u and the inner function is u = x^2 + 6.9x.

Step 2: Find the derivative of the outer function. The derivative of e^u with respect to u is e^u.

Step 3: Find the derivative of the inner function. The derivative of x^2 + 6.9x with respect to x is 2x + 6.9.

Step 4: Apply the chain rule. The derivative of the function f(x) = e^{x^2 + 6.9x} is f'(x) = e^{x^2 + 6.9x} * (2x + 6.9).

Now, to find the value of f'(0), we substitute x = 0 into the derivative function:

f'(0) = e^{(0)^2 + 6.9*(0)} * (2*(0) + 6.9) = e^0 * 6.9 = 1 * 6.9 = 6.9.

So, the value of f'(0) is 6.9.