Diffrentiate the following w.r.t.x e^ [( x) 2- x 2] - Vidyadip

Question

Diffrentiate the following w.r.t.x

$e^{\log [(\log x) 2-\log x 2]}$

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Answer

$\begin{aligned} & \text { Let } y=e^{\log [(\log x) 2-\log x 2]} \\ & =(\log x)^2-\log x^2 \ldots\left[\because e^{\log x}=x\right]\end{aligned}$

Differentiating w.r.t. x, we get

$\begin{aligned} \frac{d y}{d x} & =\frac{d}{d x}\left[(\log x)^2-2 \log x\right] \\ & =\frac{d}{d x}(\log x)^2-2 \frac{d}{d x}(\log x) \\ & =2 \log x \cdot \frac{d}{d x}(\log x)-2 \times \frac{1}{x} \\ & =2 \log x \times \frac{1}{x}-\frac{2}{x} \\ & =\frac{2 \log x}{x}-\frac{2}{x} .\end{aligned}$

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