Differentiate the following from first principles
We need to find derivative of f(x) =
Derivative of a function f(x) is given by –
f’(x) = {where h is a very small positive number}
∴ derivative of f(x) = 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, (2hx + h2) → 0
∴ multiplying numerator and denominator by (2hx + h2) in order to apply the formula –
∴ f’(x) =
Again using algebra of limits, we have –
⇒ f’(x) =
Use the formula:
⇒ f’(x) =
⇒ f’(x) =
∴ f’(x) =
Hence,
Derivative of f(x) =