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

Problem

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

Solution

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

2. Apply the Chain Rule for the natural logarithm, which states d(ln(u))/d(x)=1/u⋅d(u)/d(x)

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

4. Substitute the components into the Chain Rule formula.

d(ln(tan(x)))/d(x)=1/tan(x)⋅sec2(x)

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

1/tan(x)⋅sec2(x)=cos(x)/sin(x)⋅1/cos2(x)

6. Cancel the common factor of cos(x) in the numerator and denominator.

1/(sin(x)*cos(x))

7. Apply the double angle identity sin(2*x)=2*sin(x)*cos(x) which implies sin(x)*cos(x)=1/2*sin(2*x)

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

8. Rewrite using the cosecant identity csc(2*x)=1/sin(2*x)

2*csc(2*x)

Final Answer

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

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