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

Problem

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

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is cos(u) and the inner function is u=ln(x)

2. Apply the Chain Rule which states that the derivative of ƒ*(g(x)) is ƒ′*(g(x))⋅g(x)′

3. Differentiate the outer function with respect to the inner function. The derivative of cos(u) is −sin(u)

4. Differentiate the inner function with respect to x The derivative of ln(x) is 1/x

5. Multiply the results together to find the final derivative.

Final Answer

d(cos(ln(x)))/d(x)=−sin(ln(x))/x

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