Differentiate the following from first principles

We need to find derivative of f(x) = √(sin 2x)


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


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


derivative of f(x) = √(sin 2x) is given as –


f’(x) =


f’(x) =


We can’t evaluate the limits at this stage only as on putting value it will take 0/0 form. So, we need to do little modifications.


Multiplying numerator and denominator by √(sin 2(x + h)) + √(sin 2x), we have –


f’(x) =


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


f’(x) =


Again using algebra of limits, we get –


f’(x) =


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


f’(x) =


f’(x) =


Using algebra of limits –


f’(x) =


Use the formula –


f’(x) =


Put the value of h to evaluate the limit –


f’(x) =


Hence,


Derivative of f(x) = √(sin 2x) =


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