Find the Derivative - d/d@VAR f(x)=sin( natural log of x)

Problem

d(sin(ln(x)))/d(x)

Solution

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

2. Apply the Chain Rule by differentiating the outer function with respect to the inner function.

d(sin(u))/d(u)=cos(u)

3. Differentiate the inner function ln(x) with respect to x

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

4. Multiply the derivative of the outer function by the derivative of the inner function.

cos(ln(x))⋅1/x

5. Simplify the expression into a single fraction.

cos(ln(x))/x

Final Answer

d(sin(ln(x)))/d(x)=cos(ln(x))/x

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