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

Problem

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

Solution

1. Identify the outer function as cos(u) and the inner function as u=ln(x)

2. Apply the chain rule, which states that (d(ƒ)*(g(x)))/d(x)=ƒ′*(g(x))⋅g(x)′

3. Differentiate the outer function cos(u) with respect to u to get −sin(u)

4. Differentiate the inner function ln(x) with respect to x to get 1/x

5. Multiply the results together and substitute u=ln(x) back into the expression.

6. Simplify the resulting fraction.

Final Answer

d(cos(ln(x)))/d(x)=−sin(ln(x))/x

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