Q. 3 J5.0( 1 Vote )

Differentiate the following from first principles


Answer :

We need to find derivative of f(x) = e√(ax + b)


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


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


derivative of f(x) = e√(ax + b) is given as –


f’(x) =


f’(x) =


f’(x) =


Taking common, we have –


f’(x) =


Using algebra of limits –


f’(x) =


f’(x) =


As one of the limits can’t be evaluated by directly putting the value of h as it will take 0/0 form.


So we need to take steps to find its value.


As h 0 so, () 0


multiplying numerator and denominator by in order to apply the formula –


f’(x) =


Again using algebra of limits, we have –


f’(x) =


Use the formula:


f’(x) =


Again we get an indeterminate form, so multiplying and dividing √(ax + ah + b) + √(ax + b) 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) = e√(ax + b) =


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