Final answer:
The derivative of g(x) = e^(-x) cos(x^2) is found using the product rule and chain rule, resulting in g'(x) = -e^(-x) cos(x^2) - 2xe^(-x)sin(x^2).
Step-by-step explanation:
To find the derivative of the function g(x) = e(-x) cos(x2), we need to use the product rule and the chain rule of differentiation. The product rule states that the derivative of two functions multiplied together is the derivative of the first function times the second function plus the first function times the derivative of the second function. In this case, our first function is e(-x) and the second is cos(x2).
First, we find the derivative of e(-x), which is -e(-x) due to the chain rule, since the derivative of -x is -1. Next, we compute the derivative of cos(x2), which involves the chain rule again. The derivative of cos(u) is -sin(u), and u here is x2. Therefore, the derivative of cos(u) with respect to x is -2xsin(x2).
Combining these derivatives using the product rule, we get:
g'(x) = (-e(-x)) · cos(x2) + e(-x) · (-2xsin(x2))
Thus, the derivative of g(x) is g'(x) = -e(-x) cos(x2) - 2xe(-x)sin(x2).