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Find the derivative of a product of terms (x-ai)bi

by RoRi

Find the derivative of the following function:

    \[ f(x) = \prod_{i=1}^n (x-a_i)^{b_i}. \]

To take this derivative we want to use logarithmic differentiation. To that end we take the derivative of both sides,

    \begin{align*} \log f(x) &= \log \left( \prod_{i=1}^n (x-a_i)^{b_i} \right) \\ &= \sum_{i=1}^n b_i \log(x-a_i). \end{align*}

Therefore, taking the derivative of both sides, we have

    \begin{align*} &&\frac{f'(x)}{f(x)} &= \sum_{i=1}^n \frac{b_i}{x-a_i} \\ \implies && f'(x) &= f(x) \left( \sum_{i=1}^n \frac{b_i}{x-a_i} \right) \\ \implies && f'(x) &= \left( \prod_{i=1}^n (x-a_i)^{b_i} \right) \left( \sum_{i=1}^n \frac{b_i}{x-a_i} \right) \end{align*}

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