Find the Derivative - d/dx natural log of x^x

Problem

d(ln(xx))/d(x)

Solution

1. Apply the logarithm power rule to simplify the expression before differentiating. The rule ln(ab)=b*ln(a) allows us to move the exponent x to the front.

ln(xx)=x*ln(x)

2. Apply the product rule for differentiation, which states (d(ƒ(x))*g(x))/d(x)=ƒ(x)′*g(x)+ƒ(x)*g(x)′ Here, ƒ(x)=x and g(x)=ln(x)

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

3. Differentiate the individual components. We know d(x)/d(x)=1 and d(ln(x))/d(x)=1/x

(d(x)*ln(x))/d(x)=1⋅ln(x)+x⋅1/x

4. Simplify the resulting expression by performing the multiplication.

ln(x)+1

Final Answer

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

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