Find dy/dx y=x^x

Problem

d(y)/d(x)* for *y=xx

Solution

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

ln(y)=ln(xx)

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

ln(y)=x*ln(x)

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

d(ln(y))/d(x)=(d(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)=ln(x)d(x)/d(x)+xd(ln(x))/d(x)

5. Simplify the derivatives on the right side of the equation.

1/yd(y)/d(x)=ln(x)⋅1+x⋅1/x

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

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

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

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

d(y)/d(x)=xx*(ln(x)+1)

Final Answer

d(y)/d(x)=xx*(1+ln(x))

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