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

Problem

d(ln(cos(x)))/d(x)

Solution

1. Identify the outer and inner functions for the chain rule, where the outer function is ln(u) and the inner function is u=cos(x)

2. Apply the chain rule which states that d(ln(u))/d(x)=1/u⋅d(u)/d(x)

3. Differentiate the inner function cos(x) with respect to x to get −sin(x)

4. Substitute the components into the chain rule formula.

d(ln(cos(x)))/d(x)=1/cos(x)⋅(−sin(x))

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

d(ln(cos(x)))/d(x)=−sin(x)/cos(x)

Final Answer

d(ln(cos(x)))/d(x)=−tan(x)

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