Find the Derivative - d/dx natural log of cos(2x)

Problem

d(ln(cos(2*x)))/d(x)

Solution

1. Identify the outer function as u=ln(v) where v=cos(2*x) and apply the chain rule for natural logarithms.

d(ln(v))/d(x)=1/v⋅d(v)/d(x)

2. Differentiate the inner function v=cos(2*x) by applying the chain rule again, where the derivative of cos(w) is −sin(w)⋅d(w)/d(x)

d(cos(2*x))/d(x)=−sin(2*x)⋅(d(2)*x)/d(x)

3. Calculate the derivative of the innermost linear term 2*x

(d(2)*x)/d(x)=2

4. Combine all parts using the chain rule sequence.

d(ln(cos(2*x)))/d(x)=1/cos(2*x)⋅(−sin(2*x))⋅2

5. Simplify the expression by using the trigonometric identity tan(θ)=sin(θ)/cos(θ)

d(ln(cos(2*x)))/d(x)=−2⋅sin(2*x)/cos(2*x)

Final Answer

d(ln(cos(2*x)))/d(x)=−2*tan(2*x)

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