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

Problem

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

Solution

1. Set up the equation for logarithmic differentiation by letting y=(sin(x))ln(x)

2. Apply the natural logarithm to both sides of the equation to simplify the exponent.

ln(y)=ln((sin(x))ln(x))

3. Use the power rule for logarithms to move the exponent in front of the log.

ln(y)=ln(x)*ln(sin(x))

4. Differentiate implicitly with respect to x on both sides.

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

5. Apply the chain rule to the left side and the product rule to the right side.

1/yd(y)/d(x)=d(ln(x))/d(x)*ln(sin(x))+ln(x)d(ln(sin(x)))/d(x)

6. Compute the derivatives of the individual terms, using the chain rule for ln(sin(x))

1/yd(y)/d(x)=1/x*ln(sin(x))+ln(x)cos(x)/sin(x)

7. Simplify the expression by replacing the ratio of cosine to sine with the cotangent function.

1/yd(y)/d(x)=ln(sin(x))/x+ln(x)*cot(x)

8. Solve for the derivative d(y)/d(x) by multiplying both sides by y

d(y)/d(x)=y*(ln(sin(x))/x+ln(x)*cot(x))

9. Substitute the original expression for y back into the equation.

d(y)/d(x)=(sin(x))ln(x)*(ln(sin(x))/x+ln(x)*cot(x))

Final Answer

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

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