Find the derivative of $f (x) = 1 + x + x^{2} + x^{3} + . . . . + x^{50}$ at $x = 1$

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Solution

Given $f (x) = 1 + x + x^{2} + x . . . . x^{50}$
$f^{'} (x) = 1 + 2 x + 3 x^{2} + . . . . + 50 k^{49}$
$f^{'} (1) = 1 + 2 + 3 + . . . . .50$
$= \frac{50 (50 + 1)}{2} = 25 \times 51$
= 1275

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