Differentiate the following from first principle.

e ^{– x}

We need to find derivative of f(x) = e ^{– x}

Derivative of a function f(x) is given by –

f’(x) = {where h is a very small positive number}

∴ derivative of f(x) = e ^{– x} is given as –

f’(x) =

⇒ f’(x) =

⇒ f’(x) =

Taking e ^{– x} common, we have –

⇒ f’(x) =

Using algebra of limits –

⇒ 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.

⇒ f’(x) =

Use the formula:

⇒ f’(x) = e ^{– x} × ( – 1)

⇒ f’(x) = – e ^{– x}

Hence,

Derivative of f(x) = e ^{– x} = – e ^{– x}