Find the Derivative - d/dx sin(x)^x

Problem

d(sin(x))/d(x)

Solution

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

y=sin(x)

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

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

3. Use the power rule for logarithms to bring the exponent x down as a coefficient.

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

4. Differentiate implicitly with respect to x on both sides, using the product rule on the right side.

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

5. Apply the chain rule to differentiate ln(sin(x))

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

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

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

7. Solve for the derivative by multiplying both sides by y

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

8. Substitute back the original expression for y to get the final result in terms of x

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

Final Answer

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

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