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

Problem

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

Solution

1. Identify the outer function as the natural logarithm, ln(u) where u=cos(h(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(h(x)) using the chain rule again, noting that the derivative of cos(v) is −sin(v)

4. Calculate the derivative of the innermost function h(x) which is h(x)′

5. Combine the results using the chain rule: 1/cos(h(x))⋅(−sin(h(x)))⋅h(x)′

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

Final Answer

d(ln(cos(h(x))))/d(x)=−tan(h(x))*h(x)′

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