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)} =