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Derivative at 0

Find the derivative of

    \[f(x) = \frac{\sqrt{1+2x} \cdot \sqrt[4]{1+4x} \cdot \sqrt[6]{1+6x} \cdot \cdots \cdot \sqrt[100]{1+100x}}{\sqrt[3]{1+3x} \cdot \sqrt[5]{1+5x} \cdot \sqrt[7]{1+7x} \cdot \cdots \cdot \sqrt[101]{1+101x}}\]

at 0.

Solution

The domain of f is \mathcal{A}_f = \left [ -\frac{1}{101} , +\infty \right ). Let us now consider the logarithmic function f. Hence,

    \[g(x) = \ln f(x) = \sum_{n=2}^{101} \frac{(-1)^n}{n} \ln \left ( 1 + n x \right )\]

Differentiating we get

    \[g'(x) = \frac{f'(x)}{f(x)} = \sum_{n=2}^{101} \frac{(-1)^n}{1+nx}\]

Setting x=0 we get

    \[g'(0) = \frac{f'(0)}{f(0)} = \sum_{n=2}^{101} (-1)^n =0 \Leftrightarrow f'(0) =0\]

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