Use the limit definition of the derivative to find the derivative of f(x

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f(x)= 1 / (3x+5)

limit definition of derivative is [ f(x+h)- f(x) ] / h

May 15th, 2015

f(x+h)=1/(3(x+h)+5)

f(x+h)-f(x)=1/(3(x+h)+5)-1/(3x+5)=[3x+5 - (3(x+h)+5)]/[(3(x+h)+5)*(3x+5)]

=-3h//[(3(x+h)+5)*(3x+5)]

as h->0 this expression is of the order of h/(3x+5)^2. The h in the denominator will contribute only to terms of order h^2 and higher, and so, it is neglected

Thus:

f'(x)= lim [f(x+h)-f(x)]/h as h->0 is ={-3h//[(3x+5)*(3x+5)]}/h = -3//[(3x+5)]^2



May 15th, 2015

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