Find the derivative of sin(x)cos(x) using the product rule.When I do it I get cos^2(x)-sin^2(x), yet my calculator says it is 2(cos(x))^2 - 1.

The derivative of y = (sin x)(cos x) can be found using
the product rule.

y' = [(sin x)(cos
x)]'

=> [sin x]'[cos x] + [sin x][cos
x]'

=> [cos x][cos x] - [sin x][sin
x]

=> (cos x)^2 - (sin
x)^2

Use the relation (cos x)^2 + (sin x)^2 = 1 or (sin
x)^2 = 1 - (cos x)^2

=> (cos x)^2 - 1 + (cos
x)^2

=> 2*(cos x)^2 -
1

The result given by your calculator of
2*(cos x)^2 - 1 is equivalent to the result that you have got of (cos x)^2 - (sin
x)^2.