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

Problem

d(xln(x))/d(x)

Solution

1. Rewrite the expression using the identity ab=e(b*ln(a)) to make it easier to differentiate.

xln(x)=e(ln(x)*ln(x))

2. Simplify the exponent by combining the natural log terms.

e(ln(x)*ln(x))=e((ln(x))2)

3. Apply the chain rule for the exponential function, where d(eu)/d(x)=eu⋅d(u)/d(x)

d(e((ln(x))2))/d(x)=e((ln(x))2)⋅d(ln(x))/d(x)

4. Apply the power rule and chain rule to differentiate the exponent (ln(x))2

d(ln(x))/d(x)=2*ln(x)⋅d(ln(x))/d(x)

5. Substitute the derivative of the natural log, which is 1/x

d(ln(x))/d(x)=2*ln(x)⋅1/x

6. Combine all parts and substitute the original expression back in for e((ln(x))2)

d(xln(x))/d(x)=xln(x)⋅(2*ln(x))/x

Final Answer

d(xln(x))/d(x)=2*x(ln(x)−1)*ln(x)

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