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

Problem

d(xx)/d(x)

Solution

1. Set the function equal to y to use logarithmic differentiation.

y=xx

2. Take the natural log of both sides to move the exponent.

ln(y)=ln(xx)

3. Apply the power rule for logarithms to simplify the right side.

ln(y)=x*ln(x)

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

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

5. Apply the chain rule on the left and the product rule on the right.

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

6. Simplify the derivatives on the right side.

1/yd(y)/d(x)=ln(x)⋅1+x⋅1/x

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

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

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

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

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

Final Answer

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

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