Q70 / 125 · BITSAT 2011 · Mathematics

If \(y=x^{x^2}\), then \(\frac{d y}{d x}\) is equal to

  • A
    \((2 \ln x)\)
  • B
    \((2 \ln x+1)\)
  • C
    \((\ln \ln x+1) x^{x^2}\)
  • Correct answer
    D
    None of these

Solution

Consider \(y=x^{x^2} \Rightarrow \ln y=x^2 \ln x\) \(\begin{aligned} & \frac{1}{y} \frac{d y}{d x}=2 x \ln x+x^2 \cdot \frac{1}{x}=x(1+2 \ln x) \\ & \frac{d y}{d x}=x^{x^2} \cdot x(1+2 \ln x)=x^{x^2+1}(1+2 \ln x) \end{aligned}\)

Hence, the correct answer is option: 'D'.

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