Question

Evaluate d/dx x^x ln(x)

(a) x^(x-1) + x^2x ln⁡(x) + x^x [ln⁡(x)]^2

(b) x^(x-1) + x^x ln⁡(x) + x^x [ln⁡(x)]^2

(c) x^(x-1) + x^x ln⁡(x) + x^x ln⁡(x)

(d) x^x + x^x ln⁡(x) + x^x [ln⁡(x)]^2

The question was asked in a job interview.

This key question is from Limits and Derivatives of Several Variables in chapter Partial Differentiation of Engineering Mathematics

Answer 1

The correct option is (b) x^(x-1) + x^x ln⁡(x) + x^x [ln⁡(x)]^2

Easiest explanation: \(\frac{dx^x ln⁡(x)}{dx}=\frac{x^x dln(x)}{dx}+\frac{ln⁡(x)dx^x}{dx}\)

Now

Y = x^x

Taking log on both side.

ln(y) = xln(x)

Differntiating both sides

\(\frac{1}{y} \frac{dy}{dx}=1+ln⁡(x)\)

\(\frac{dy}{dx}\)=x^x (1+ln⁡(x))

Hence,

\(\frac{dx^x ln⁡(x)}{dx}=x^{x-1}+x^x ln⁡(x) (1+ln⁡(x) )=x^{x-1}+x^x ln⁡(x)+x^x [ln⁡(x)]^2\)