Q. 5 A

# Differentiate the following from first principles

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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