Find the Derivative - d/dx natural log of cos(3x)

Problem

d(ln(cos(3*x)))/d(x)

Solution

1. Identify the outer function as ln(u) where u=cos(3*x)

2. Apply the chain rule for the natural logarithm, which states d(ln(u))/d(x)=1/u⋅d(u)/d(x)

3. Differentiate the inner function u=cos(3*x) using the chain rule again, where the derivative of cos(v) is −sin(v)⋅d(v)/d(x)

4. Calculate the derivative of the innermost argument 3*x which is 3

5. Combine the results of the chain rule steps.

d(ln(cos(3*x)))/d(x)=1/cos(3*x)⋅(−sin(3*x))⋅3

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

d(ln(cos(3*x)))/d(x)=−3*tan(3*x)

Final Answer

d(ln(cos(3*x)))/d(x)=−3*tan(3*x)

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