Find dy/dx y=sin(x)^x

Problem

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

Solution

1. Apply logarithmic differentiation by taking the natural logarithm of both sides to handle the variable in the exponent.

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

2. Use the power rule for logarithms to move the exponent x in front of the natural log.

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

3. Differentiate both sides with respect to x using the chain rule on the left side and the product rule on the right side.

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

4. Apply the product rule d()/d(x)*[u*v]=u′*v+u*v′ to the right side.

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

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

1/yd(y)/d(x)=ln(sin(x))+x1/sin(x)*cos(x)

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

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

7. Solve for dy/dx by multiplying both sides by y

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

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

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

Final Answer

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

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