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Multiple Choice
Find the derivative of the function: .
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Verified step by step guidance
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Step 1: Recognize that the function y = e^{tan(x)} is a composition of functions. Specifically, it involves the exponential function e^u, where u = tan(x). To differentiate this, we will use the chain rule.
Step 2: Recall the chain rule: If y = f(g(x)), then y' = f'(g(x)) * g'(x). Here, f(u) = e^u and g(x) = tan(x).
Step 3: Differentiate the outer function f(u) = e^u with respect to u. The derivative of e^u is e^u.
Step 4: Differentiate the inner function g(x) = tan(x) with respect to x. The derivative of tan(x) is sec^2(x).
Step 5: Combine the results using the chain rule: y' = e^{tan(x)} * sec^2(x). This is the derivative of the given function.