To find the derivative of the function, we can use the product rule and chain rule.
Let's break down the function into two parts:
1. y = cos(x)
2. g(x) = ln(cos^2(x))
Now, let's find the derivative of each part individually:
1. For y = cos(x), the derivative is dy/dx = -sin(x)
2. For g(x) = ln(cos^2(x)), we can use the chain rule. Let u = cos^2(x), then g(x) = ln(u), and the derivative is dg/du * du/dx:
dg/du = 1/u
du/dx = -2sin(x)cos(x) (using the chain rule on u = cos^2(x))
So, the derivative of g(x) with respect to x is dg/dx = dg/du * du/dx = (1/u) * (-2sin(x)cos(x)) = -2sin(x)cos(x) * (1/cos^2(x)) = -2tan(x)
Now, using the product rule, the derivative of the function y = cos(x) * g(x) is:
dy/dx = (cos(x) * (-2tan(x))) + (-sin(x) * g(x))
Simplifying this expression, we get:
dy/dx = -2sin(x)tan(x) - sin(x)ln(cos^2(x))
Therefore, the derivative of the function y = cos(x) * ln(cos^2(x)) is dy/dx = -2sin(x)tan(x) - sin(x)ln(cos^2(x)).
Now that we have the derivative, we can proceed to the next part of the question.