Lesson Find the derivative of a function defined by complicated expression

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Find the derivative of a function defined by complicated expression


Problem 1

If   y = %281+%2B+x%29%2A%281+%2B+x%5E2%29%2A%281+%2B+x%5E3%29%2Aellipsis%2A%281+%2B+x%5En%29,   find   %28dy%29%2F%28dx%29   at x = 1.

Solution 

Apply the formula for the derivative of a product.


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


The k-th addend is

    .


At x= 1, the k-th addend is  %28%282%5En%29%2F2%29%2Ak = k%2A2%5E%28n-1%29.


The sum of n these addends is  2%5E%28n-1%29%2A%28%28n%2A%28n%2B1%29%29%2F2%29 = 2%5E%28n-2%29%2An%2A%28n%2B1%29.    ANSWER


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