Question

Find the derivative of f(x) = -ln(2cos(x)).

Answer 1

The derivative of the function f(x) = -ln(2cos(x)) is found using the chain rule and simplifies to f'(x) = tan(x).

Step-by-step explanation:

To find the derivative of f(x) = -ln(2cos(x)), we can utilize the chain rule for differentiation and the rule that the derivative of ln(x) is 1/x. The chain rule tells us that the derivative of a composed function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

For f(x) = -ln(2cos(x)), here's how we apply the chain rule step by step:

1. Identify the outer function as -ln(u), where u = 2cos(x), and the inner function as u = 2cos(x).
2. The derivative of -ln(u) with respect to u is -1/u.
3. The derivative of u = 2cos(x) with respect to x is -2sin(x) because the derivative of cos(x) is -sin(x).
4. Multiply the derivatives from steps 2 and 3 together.

The derivative of f(x) = -ln(2cos(x)) is therefore f'(x) = (-1/2cos(x)) * (-2sin(x)) which simplifies to f'(x) = sin(x)/cos(x).

By recognizing that sin(x)/cos(x) is equivalent to tan(x), we can simplify further:

Therefore, the final derivative of f(x) is f'(x) = tan(x).