Differentiate ln(cos(2x))

just use the chain rule

d/dx ln(u) = 1/u du/dx

d/dx ln(cos(2x)) = 1/cos(2x) (-2sin(2x)) = -2tan(2x)

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To differentiate ln(cos(2x)), we will use the chain rule. The chain rule states that if we have a composition of functions, such as f(g(x)), then the derivative of the composition is given by the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Let's break down the problem step by step:

Step 1: Identify the outer and inner functions.
In this case, the outer function is ln(x) and the inner function is cos(2x).

Step 2: Differentiate the outer function.
The derivative of ln(x) is 1/x. So, the derivative of the outer function ln(cos(2x)) is 1/cos(2x).

Step 3: Differentiate the inner function.
The derivative of cos(2x) is -2sin(2x). Note that we have multiplied by the derivative of the inner function as per the chain rule.

Step 4: Multiply the derivatives.
Now, multiply the derivatives of the outer and inner functions: (1/cos(2x)) * (-2sin(2x)).

Step 5: Simplify the expression.
To simplify, you can multiply the terms together and combine like terms if possible. However, keep in mind that some trigonometric identities might be useful in simplifying further.

So, the final derivative of ln(cos(2x)) is (-2sin(2x))/(cos(2x)).

Remember to always check for the validity of your answer and apply any simplifications if necessary.