Explanation:
To find the derivative of the expression sin(4x)cos(x), you can use the product rule, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
Let u = sin(4x) and v = cos(x). Then, we have:
u' = d/dx[sin(4x)] = 4cos(4x) (using the chain rule)
v' = d/dx[cos(x)] = -sin(x)
Now, apply the product rule:
(uv)' = u'v + uv'
= (4cos(4x))(cos(x)) + (sin(4x))(-sin(x))
= 4cos(4x)cos(x) - sin(4x)sin(x)
So, the derivative of sin(4x)cos(x) is:
4cos(4x)cos(x) - sin(4x)sin(x)