How do you differentiate the function sin²(x) cos(x)?

To differentiate the function sin²(x) cos(x), we will use the product rule as well as the chain rule. The product rule states that if you have two functions, u(x) and v(x), the derivative of their product is given by:

 (u v)' = u' v + u v'

In our case, let:

• u = sin²(x)
• v = cos(x)

We now need to find the derivatives of u and v:

1. Finding u’ (the derivative of sin²(x)):

We will use the chain rule here:

u = sin²(x) = (sin(x))^2

Let w = sin(x), then:

u = w²

2. Now, using the chain rule:

u' = 2w rac{dw}{dx} = 2 sin(x) cos(x)

3. So:

u' = 2 sin(x) cos(x)

Now, let’s find v’:

v = cos(x)

v' = -sin(x)

Now we have:

• u’ = 2 sin(x) cos(x)
• v’ = -sin(x)

Now, applying the product rule:

(u v)' = u' v + u v'

Substituting in our values:

(sin²(x) cos(x))' = (2 sin(x) cos(x)) cos(x) + sin²(x) (-sin(x))

This simplifies to:

2 sin(x) cos²(x) - sin²(x) sin(x)

Thus, the final derivative of the function sin²(x) cos(x) is:

2 sin(x) cos²(x) - sin³(x)