# Derivative of *e*^{x}

How to solve the derivative of *e*^{x} problems: formula, proof, examples, and their solutions.

## Formula

The derivative of *e*^{x} is itself: *e*^{x}.

## Proof

ln both sides.

Natural logarithms

Differentiate both sides.

Implicit differentiation

Derivative of ln *x*

Multiply *y* on both sides.

Change *y* into *e*^{x}.

So [*e*^{x}]' = *e*^{x}.

## Example 1

*y* = (*x*^{2})(*e*^{x})

So *y*' is equal to,

the derivative of *x*^{2}, 2*x*

times *e*^{x}

plus *x*^{2},

times, the derivative of *e*^{x}, *e*^{x}.

Product rule in differentiation

Power rule in differentiation (Part 1)

## Example 2

*y* = *e*^{x2 - 4x}

So *y*' is equal to,

the derivative of the outer part, *e*^{x2 - 4x}

times, the derivative of the inner part, (2*x* - 4).

Chain rule in differentiation

Derivative of polynomials