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Derivative at 0

Let

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

Evaluate f'(0).

Solution

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

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

and differentiate it; Hence,

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

For x=0 we have

    \[g'(0) = \sum_{k=2}^{101} (-1)^k =0\]

Thus, f'(0)=0 since f(0)=1.

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