Final answer:
To find the derivative of y = tan(e^x), apply the chain rule, resulting in the derivative sec^2(e^x) * e^x.
Step-by-step explanation:
The student asked how to calculate the derivative of the function y = tan(e^x). To find this derivative, we'll use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function.
The outer function is tan(u), where u is the inner function e^x. The derivative of tan(u) with respect to u is sec^2(u). The inner function e^x has a derivative with respect to x, which is simply e^x.
Applying the chain rule, we get:
d/dx [tan(e^x)] = sec^2(e^x) * d/dx [e^x] = sec^2(e^x) * e^x.
That is the derivative of the given function.