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Find the derivative of x * ln(x)

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Answer
ln⁡(x)+1\boxed{\ln(x) + 1}

Checking the result by integrating the derivative ∫(ln⁡(x)+1) dx=xln⁡(x)−x+x=xln⁡(x)\int (\ln(x) + 1) \, dx = x \ln(x) - x + x = x \ln(x), which matches our original function.

Problem: Find the derivative of xln⁡(x)x \ln(x), i.e., ddx[xln⁡(x)]\frac{d}{dx} [x \ln(x)].

Steps

  1. Apply the product rule
    ddx[x]⋅ln⁡(x)+x⋅ddx[ln⁡(x)]\frac{d}{dx}[x] \cdot \ln(x) + x \cdot \frac{d}{dx}[\ln(x)]
  2. Differentiate each term
    1⋅ln⁡(x)+x⋅1x1 \cdot \ln(x) + x \cdot \frac{1}{x}
  3. Simplify the expression
    ln⁡(x)+1\ln(x) + 1

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