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)