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The monotony of a function shows the behavior of the
function: increasing or decreasing function.

A function is
strictly increasing if it's first derivative is positive and it is decreasing if it's
first derivative is negative.

We'll re-write the function,
based on the fact that the sine function is odd:

f(x) =
xcos x - sin x

We'll calculate the first
derivative:

f'(x)= (xcos x – sin
x)'

f'(x) = ( xcos x)'-( sin
x)'

We notice that the first term is a product, so we'll
apply the product rule:

f'(x) = x'*(cos x)+x*(cos x)' – cos
x

f'(x)=1*cos x-x*sin x –cos
x

We'll eliminate like
terms:

f'(x)= -x*sin x

Since
the sine function is positive over the range [0 ; 180], the values of x are positive and
the product is negative, the first derivative is
negative.

The function y = f(x) =
xcosx+sin(-x) is decreasing over the range [0,
pi].