Find the Derivative - d/dx y=x^(tan(x))

Problem

d()/d(x)*xtan(x)

Solution

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

ln(y)=ln(xtan(x))

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

ln(y)=tan(x)*ln(x)

3. Differentiate implicitly with respect to x on both sides of the equation.

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

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

1/yd(y)/d(x)=sec2(x)*ln(x)+tan(x)1/x

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

d(y)/d(x)=y*(sec2(x)*ln(x)+tan(x)/x)

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

d(y)/d(x)=xtan(x)*(sec2(x)*ln(x)+tan(x)/x)

Final Answer

d(xtan(x))/d(x)=xtan(x)*(sec2(x)*ln(x)+tan(x)/x)

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