What is the derivative of k(x)=sin x cos x?

A. -sin x cos x
B. -2 sin x cos x
C. 2 cos^2x-1
D. sin^2x - cos^2x

k(x)=sin x cos x

think:
k(x)= (sin x)(cos x) , and use the product rule

Use the product rule,

d(sin(x)cos(x))/dx
=cos(x)dsin(x)/dx+sin(x)dcos(x)/dx
=cos²(x)-sin²(x)

Next use sin²(x)+cos²(x)=1 to transform the above result to one of the provided answers.

Note:
an easier way to derive the given expression is to make use of the identity:
sin(2x)=2sin(x)cos(x)
so
d(sin(2x))/dx
=d((1/2)sin(2x))/dx
=cos(2x)
(which equals cos²(x)-sin²(x) AND one of the posted answers)

To find the derivative of k(x) = sin(x) cos(x), we can use the product rule.

The product rule states that if you have two functions u(x) and v(x), where u(x) and v(x) are both differentiable, then the derivative of their product is given by:

(d/dx) (u(x) v(x)) = u'(x) v(x) + u(x) v'(x)

In this case, u(x) = sin(x) and v(x) = cos(x). We can find the derivatives of these functions using the trigonometric derivative rules:

(d/dx) sin(x) = cos(x)
(d/dx) cos(x) = -sin(x)

Applying the product rule, we have:

(d/dx) (sin(x) cos(x)) = (cos(x) cos(x)) + (sin(x) (-sin(x)))
= cos^2(x) - sin^2(x)

Therefore, the derivative of k(x) = sin(x) cos(x) is sin^2(x) - cos^2(x).

So the correct answer is D. sin^2x - cos^2x.