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

Problem

d(ln(tan(x)))/d(x)

Solution

1. Identify the outer and inner functions to apply the chain rule for the natural logarithm.

2. Apply the rule for the derivative of ln(u) which is 1/u⋅d(u)/d(x)

3. Differentiate the inner function tan(x) which results in sec2(x)

4. Substitute these parts into the chain rule formula:

d(ln(tan(x)))/d(x)=1/tan(x)⋅sec2(x)

5. Simplify the expression using trigonometric identities where 1/tan(x)=cos(x)/sin(x) and sec2(x)=1/cos2(x)

cos(x)/sin(x)⋅1/cos2(x)=1/(sin(x)*cos(x))

6. Apply the double angle identity sin(2*x)=2*sin(x)*cos(x) to further simplify if desired, or use reciprocal identities.

1/(sin(x)*cos(x))=2/(2*sin(x)*cos(x))=2/sin(2*x)=2*csc(2*x)

Final Answer

d(ln(tan(x)))/d(x)=2*csc(2*x)

Want more problems? Check here!