Differentiate each of the following from first principles:

We need to find derivative of f(x) = 1/x3


Derivative of a function f(x) from first principle is given by –


{where h is a very small positive number}


derivative of f(x) = 1/x3 is given as –





Using algebra of limits –




Using a3 – b3 = (a – b)(a2 + ab + b2)


We have:




As h is cancelled and by putting h = 0 we are not getting any indeterminate form so we can evaluate the limit directly.






Hence,


Derivative of f(x) = 1/x3


1