Question 1200001
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If x = (1 + y) (1 + y²) (1 + y³)... (1 + y^n)
Find dx/dy (when y = 1)
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<pre>
Apply the formula for the derivative of a product.


You will get that the derivative dx/dy is the sum of n addends.


The k-th addend is

    {{{(((1 + y)*(1 + y^2)*(1 + y^3)* ellipsis*(1 + y^n))/(1+y^k))*(k*y^(k-1))}}}.


At y= 1, the k-th addend is  {{{((2^n)/2)*k}}} = {{{k*2^(n-1)}}}.


The sum of n these addends is  {{{2^(n-1)*((n*(n+1))/2)}}} = {{{2^(n-2)*n*(n+1)}}}.    <U>ANSWER</U>
</pre>

Solved.