What is the result of the derivative d/dx of sin^2(x)?

To find the derivative of sin^2(x) with respect to x, we can apply the chain rule. The chain rule states that if you have a composite function, say f(g(x)), the derivative is f'(g(x)) * g'(x).

In this case, we can consider sin^2(x) as (sin(x))^2. Here, the outer function is f(u) = u^2 where u = sin(x), and the inner function is g(x) = sin(x).

First, we differentiate the outer function f(u) = u^2, which gives us:

f'(u) = 2u.

Next, we need to find the derivative of the inner function:

g'(x) = cos(x).

Now, applying the chain rule, we multiply the derivative of the outer function evaluated at the inner function by the derivative of the inner function:

d/dx [sin^2(x)] = f'(g(x)) * g'(x) = 2(sin(x)) * cos(x).

This simplifies to:

2sin(x)cos(x).

This result matches the first choice. Hence, it correctly shows the derivative of sin^2(x), reflecting the application of fundamental calculus concepts.