Q. 5 A

Differentiate the following from first principles


Answer :

We need to find derivative of f(x) =


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) = is given as –


f’(x) =


f’(x) =


Use: sin A – sin B = 2 cos ((A + B)/2) sin ((A – B)/2)


f’(x) =


Using algebra of limits, we have –


f’(x) = 2


f’(x) = 2


f’(x) =


As, h 0 0


To use the sandwich theorem to evaluate the limit, we need in denominator. So multiplying this in numerator and denominator.


f’(x) =


Using algebra of limits –


f’(x) =


Use the formula:


f’(x) = × 1 ×


f’(x) =


f’(x) =


Again we get an indeterminate form, so multiplying and dividing √(2x + 2h) + √(2x) to get rid of indeterminate form.


f’(x) =


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


f’(x) =


Using algebra of limits we have –


f’(x) =


f’(x) =


f’(x) =


f’(x) =


Hence,


Derivative of f(x) = sin √2x =


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