Differentiate each of the following from first principles:

We need to find derivative of f(x) = √(2x2 + 1)


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


f’(x) = {where h is a very small positive number}


derivative of f(x) = √(2x2 + 1) is given as –


f’(x) =


f’(x) =


As the above limit can’t be evaluated by putting the value of h because it takes 0/0 (indeterminate form)


multiplying denominator and numerator by to eliminate the indeterminate form.


f’(x) =


Using algebra of limits & a2 – b2 = (a + b)(a – b),we have –


f’(x) =


f’(x) =


f’(x) =


Using a2 – b2 = (a + b)(a – b), we have –


f’(x) =


f’(x) =


f’(x) =


Evaluating the limit by putting h = 0


f’(x) =


f’(x) =


Hence,


Derivative of f(x) = √(2x2 + 1)


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