To differentiate the function sin²(x) cos(x), we will apply the product rule and the chain rule of differentiation.
The product rule states that if you have two functions multiplied together, say u and v, then the derivative is given by:
- (u * v)’ = u’ * v + u * v’
In our case, let:
- u = sin²(x)
- v = cos(x)
Now, we need to find the derivatives of u and v:
- For u: Since u = sin²(x), using the chain rule, we have:
- u’ = 2sin(x) * cos(x) = sin(2x) (using the double angle formula)
- For v: The derivative of cos(x) is:
- v’ = -sin(x)
Now, apply the product rule:
- (sin²(x) cos(x))’ = u’ * v + u * v’
- = (sin(2x)) * (cos(x)) + (sin²(x)) * (-sin(x))
Putting this all together, we get:
- (sin²(x) cos(x))’ = sin(2x) cos(x) – sin³(x)
Thus, the derivative of the function sin²(x) cos(x) is:
- sin(2x) cos(x) – sin³(x)