Differentiate: y = xsin(x)

This is a function which is in the form, 

y = f(x)g(x)

It's the product of two functions and so we must make use of the product rule. This is a simple formula which you have to remember:

dy/dx = f'(x)g(x) + f(x)g'(x).

In words: the derivative of first function multiplied by the original second function, plus, the derivative of the second function multiplied by the original first function.

In this question,

f(x) = x

g(x) = sin(x)

so we can find that, 

f'(x) = 1

g'(x) = cos(x)

and by substituting this into the formula for the product rule we get the answer:

dy/dx = sin(x) + xcos(x).