Differentiate xcos(x) with respect to x.

How do we know which method of differentiation to use in this example?
Well in ‘xos(x)’, we have 2 different functions: ‘x’ and ‘cos(x)’. Therefore, we must differentiate using the product rule.

What is the product rule and how do I use it in this case?
The product rule states that for y=f(x)g(x), dy/dx = f’(x)g(x) + g’(x)f(x) where f’(x) is the first differential of f(x) with respect to x and g’(x) is the first differential of g(x) with respect to x - this sounds complicated but let’s break it down.

For any question given, make y=function given (so y=xcos(x) here). In this case f(x) = x and g(x) = cos(x).

Please note it does not matter if we make f(x)=x and g(x)=cos(x) or if we make f(x)=cos(x) and g(x)=x - we will get the same answer regardless.

Since f(x)=x, f’(x)=1. Since g(x)=cos(x), g’(x)=-sin(x) as -sin(x) is the differential of cos(x) with respect to x.

Finally, using dy/dx = f’(x)g(x) + g’(x)f(x), dy/dx = (1)(cos(x)) + (x)(-sin(x)) = cos(x)-xsin(x).

Note that putting 2 brackets next to each other means multiplying them, for example (2)(x) = 2*x = 2x.