Find the Derivative - d/dx y=x^x

Problem

d()/d(x)*xx

Solution

1. Rewrite the function using the identity xx=eln(xx) to prepare for differentiation using the chain rule.

y=e(x*ln(x))

2. Apply the chain rule by differentiating the outer exponential function and multiplying by the derivative of the exponent.

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

3. Apply the product rule to the exponent x*ln(x) where the derivative of u⋅v is u′*v+u*v′

(d(x)*ln(x))/d(x)=1⋅ln(x)+x⋅1/x

4. Simplify the result of the product rule.

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

5. Substitute the simplified derivative and the original expression for e(x*ln(x)) 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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