Find the Derivative - d/d@VAR f(x) = natural log of sin(x)^2

Problem

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

Solution

1. Identify the function as a composition of functions, which requires the use of the chain rule.

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

3. Substitute u=sin(x) into the derivative formula.

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

4. Apply the power rule and the chain rule again to find the derivative of sin(x)

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

5. Differentiate the trigonometric function sin(x) to get cos(x)

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

6. Combine the results back into the main expression.

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

7. Simplify the expression by canceling one sin(x) term and using the identity cos(x)/sin(x)=cot(x)

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

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

Final Answer

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

Want more problems? Check here!