Differentiate the following from first principles
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) =