Final answer:
To find f'(3) for the function f(x) = 20e^x - e^(x^e), differentiate each term separately and then evaluate at x = 3 using the standard rules of differentiation and the chain rule.
Step-by-step explanation:
The question is asking to find the first derivative of the function f(x) = 20e^x - e^(x^e) at x = 3, denoted as f'(3).
To solve this, we need to apply the rules of differentiation. The derivative of e^x with respect to x is simply e^x, while for e^(x^e), we have to apply the chain rule: the derivative of e^(g(x)) is e^(g(x))g'(x), where g(x) = x^e. Since the derivative of x^e with respect to x is ex^(e-1), we multiply this by e^(x^e).
Therefore, the derivative f'(x) is 20e^x - ex^(e-1)e^(x^e). To find f'(3), we would substitute x with 3 in this expression and compute the result. This will give us the rate of change of the function f(x) at x = 3.