F(x)= ln (cos theta)

f'(x)

To find the derivative of the function f(x) = ln(cos θ), we need to apply the chain rule since we have a composition of functions (natural logarithm and cosine).

Let's break down the process step by step:

Step 1: Determine the inner function and apply its derivative.
In this case, the inner function is cos θ. The derivative of cos θ with respect to θ is -sin θ.

Step 2: Determine the outer function and apply its derivative.
The outer function is ln u, where u is the inner function cos θ. The derivative of ln u with respect to u is 1/u.

Step 3: Apply the chain rule.
The chain rule states that if you have a composition of functions f(g(x)), then the derivative is given by (f'(g(x))) * (g'(x)). In this case, f(x) = ln u and g(x) = cos θ, so the derivative can be calculated as follows:

f'(x) = (1/u) * (-sin θ)

Finally, substitute u = cos θ into the equation:

f'(x) = (1/cos θ) * (-sin θ)
f'(x) = -sin θ / cos θ

The derivative of f(x) = ln(cos θ) with respect to θ is -sin θ / cos θ.