Find the Derivative - d/dx natural log of sin(x)^2

Problem

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

Solution

1. Apply the property of logarithms to simplify the expression before differentiating. Use the rule ln(ab)=b*ln(a)

ln(sin(x))=2*ln(sin(x))

2. Apply the constant multiple rule to move the constant outside the derivative.

(d(2)*ln(sin(x)))/d(x)=2d(ln(sin(x)))/d(x)

3. Apply the chain rule for the natural logarithm, where d(ln(u))/d(x)=1/u⋅d(u)/d(x) Here, u=sin(x)

2d(ln(sin(x)))/d(x)=2⋅1/sin(x)⋅d(sin(x))/d(x)

4. Differentiate the inner function sin(x) which results in cos(x)

2⋅1/sin(x)⋅cos(x)=(2*cos(x))/sin(x)

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

(2*cos(x))/sin(x)=2*cot(x)

Final Answer

d(ln(sin(x)))/d(x)=2*cot(x)

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