Final answer:
The student's question involves finding the derivative of a composite function at a specific point using the product rule and chain rule of differentiation.
Step-by-step explanation:
The student has asked for the derivative of the function f(x) = 2(sin(x))x evaluated at x = 1. To find f'(1), we need to apply the product rule along with the chain rule of differentiation since the function involves both a power function of x and the sine function. Let's denote u(x) = 2 and v(x) = (sin(x))x. Then, by product rule, f'(x) = u'(x)v(x) + u(x)v'(x). Since deriving v(x) involves the differentiation of an exponential function with a variable in both the base and the exponent, we also need to apply the chain rule. The step-by-step differentiation would involve logarithmic differentiation for the term v(x). Once we have obtained f'(x), we substitute x = 1 to find f'(1).