Find the Derivative - d/dx y=x^( natural log of x)

Problem

d()/d(x)*x^ln(x)

Solution

1. Set up the equation for logarithmic differentiation by letting y=x^ln(x)

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

ln(y)=ln(x^ln(x))

3. Apply the power rule for logarithms, which states ln(a^b)=b*ln(a)

ln(y)=ln(x)⋅ln(x)

ln(y)=(ln(x))^2

4. Differentiate implicitly both sides with respect to x

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

5. Apply the chain rule to both sides.

1/yd(y)/d(x)=2*ln(x)⋅d(ln(x))/d(x)

1/yd(y)/d(x)=2*ln(x)⋅1/x

6. Solve for the derivative d(y)/d(x) by multiplying both sides by y

d(y)/d(x)=y⋅(2*ln(x))/x

7. Substitute back the original expression for y which is x^ln(x)

d(y)/d(x)=x^ln(x)⋅(2*ln(x))/x

8. Simplify the expression using the properties of exponents, specifically (x^a)/(x^1)=x^(a−1)

d(y)/d(x)=2*ln(x)*x^(ln(x)−1)

Final Answer

d(x^ln(x))/d(x)=2*ln(x)*x^(ln(x)−1)

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