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

Question

If $y=x^{x^2}$, then $\frac{d y}{d x}$ is equal to, $(2 \ln x)$, $(2 \ln x+1)$, $(\ln \ln x+1) x^{x^2}$, None of these

MARKS App · Accepted Answer

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

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