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

Problem

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

Solution

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

2. Apply the chain rule by taking the derivative of the natural log, which is 1/u and multiplying it by the derivative of the inner function cos(x)

d(ln(cos(x)))/d(x)=1/cos(x)⋅d(cos(x))/d(x)

3. Differentiate the inner function cos(x) which results in −sin(x)

d(ln(cos(x)))/d(x)=1/cos(x)⋅(−sin(x))

4. Simplify the expression by combining the terms into a single fraction.

d(ln(cos(x)))/d(x)=−sin(x)/cos(x)

5. Use the trigonometric identity tan(x)=sin(x)/cos(x) to write the final result.

d(ln(cos(x)))/d(x)=−tan(x)

Final Answer

d(ln(cos(x)))/d(x)=−tan(x)

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