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

Problem

d(ln(cot(x)))/d(x)

Solution

1. Identify the outer and inner functions to apply the chain rule for the natural logarithm, where the derivative of ln(u) is 1/u⋅d(u)/d(x)

2. Apply the chain rule by taking the reciprocal of the inner function cot(x) and multiplying it by the derivative of cot(x)

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

3. Differentiate the inner function cot(x) which results in −csc2(x)

d(ln(cot(x)))/d(x)=1/cot(x)⋅(−csc2(x))

4. Simplify the expression using trigonometric identities cot(x)=cos(x)/sin(x) and csc(x)=1/sin(x)

d(ln(cot(x)))/d(x)=sin(x)/cos(x)⋅(−1/sin2(x))

5. Cancel the common factor of sin(x) in the numerator and denominator.

d(ln(cot(x)))/d(x)=−1/(cos(x)*sin(x))

6. Apply the double angle identity sin(2*x)=2*sin(x)*cos(x) or simply express the result in terms of reciprocal functions sec(x) and csc(x)

d(ln(cot(x)))/d(x)=−sec(x)*csc(x)

Final Answer

d(ln(cot(x)))/d(x)=−sec(x)*csc(x)

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