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

Problem

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

Solution

1. Identify the outer function as ƒ(u)=ln(u) and the inner function as g(x)=sin(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 ln(u) with respect to u to get 1/u

4. Substitute u=sin(x) into the derivative of the outer function to get 1/sin(x)

5. Differentiate the inner function sin(x) with respect to x to get cos(x)

6. Multiply the results of the derivatives together.

d(ln(sin(x)))/d(x)=1/sin(x)⋅cos(x)

7. Simplify the expression using the trigonometric identity cot(x)=cos(x)/sin(x)

cos(x)/sin(x)=cot(x)

Final Answer

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

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