Find the Derivative - d/dx natural log of sin(5x)

Problem

d(ln(sin(5*x)))/d(x)

Solution

1. Identify the outer function as ln(u) and the inner function as u=sin(5*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 sin(5*x) using the chain rule again, where the derivative of sin(v) is cos(v)⋅d(v)/d(x)

4. Calculate the derivative of the innermost argument 5*x which is 5

5. Combine the results of the chain rule steps.

d(ln(sin(5*x)))/d(x)=1/sin(5*x)⋅cos(5*x)⋅5

6. Simplify the expression using the trigonometric identity cos(θ)/sin(θ)=cot(θ)

(5*cos(5*x))/sin(5*x)=5*cot(5*x)

Final Answer

d(ln(sin(5*x)))/d(x)=5*cot(5*x)

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