Final Answer:
The derivative of f(x) = arctan(e²x) at the point where x = 0 is 2.
Step-by-step explanation:
To find the derivative of f(x) = arctan(e²x), we apply the chain rule. Let u = e²x, then f(x) = arctan(u). The chain rule states that if g(u) = arctan(u), then g'(u) = 1 / (1 + u²). Now, applying the chain rule, we get f'(x) = g'(u) * u', where u' = du/dx = 2e²x. Therefore, f'(x) = 2e²x / (1 + e⁴x).
To find the derivative at the point x = 0, substitute x = 0 into the derivative expression. This gives f'(0) = 2e⁰ / (1 + e⁰) = 2. Thus, the derivative of f(x) at the point x = 0 is 2.
Understanding the chain rule and how to apply it in this context is essential in calculus. The chain rule is a fundamental concept that allows us to find the derivative of composite functions. In this case, recognizing that arctan(e²x) can be treated as a composite function helps simplify the differentiation process. The final result, f'(x) = 2e²x / (1 + e⁴x), provides the derivative of f(x), and evaluating it at x = 0 gives the specific value of 2.