the THIRD trouble
consider the function y=1/[x+x^{2}]
find the value of 100000 * + 9999900000 * as value of x tends to zero.
denotes the n'th derivative of y.
The answer is 0.
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1 solution
given y=1/[x+x^{2}] hence y(x+x^{2}) = 1. (asterix denotes multiplication)
differentiating once we get
y ( 1 ) * (x+x^{2}) + y * (2x+1) =0 ( first derivative relation )
differentiating again we get ....
y ( 2 ) * (x+x^{2}) + 2 * y ( 1 ) (2x+1) + 2 * y =0 ( second derivative relation )
going on like that ... we get the general form to be ........
(x+x^{2}) * y ( n ) + (2x+1) * y ( n − 1 ) * n + n * (n-1) * y ( n − 2 ) = 0
we need the derivative at x=0 ........... plugging in x=0 and n=100000 we get
100000 * y ( 9 9 9 9 9 ) + 9999900000 * y ( 9 9 9 9 8 ) = 0. hence '0' is the corect answer. *i love zero :) *
(NOTE: you may solve the previous problems named 'troubles' in the same manner .... but they are not that straight forward. This trick is very powerful and can be used in appropriate contexts)