# Derivative of e^{x}

How to find the derivative of the given function by using the derivative of e^{x}: formula, 2 examples, and their solutions.

## Formula

### Formula

The derivative of e^{x}, [e^{x}]',

is itself: e^{x}.

## Example 1

### Example

### Solution

x^{2}e^{x} is the product of x^{2} and e^{x}.

So the derivative of the product is,

the derivative of x^{2}, 2x^{1}

times e^{x}

plus

x^{2}

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

2x^{1}⋅e^{x} + x^{2}⋅e^{x} = e^{x}(x^{2} + 2x).

Common Monomial Factor

e^{x}(x^{2} + 2x) is the derivative of x^{2}e^{x}.

## Example 2

### Example

### Solution

y = e^{x2 - 4x} is a composite function of

y = e^{(whole)} and (whole) = x^{2} - 4x.

So the derivative of the composite function is,

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

times,

the derivative of the inner function x^{2} - 4x, 2x^{1} - 4.

e^{x2 - 4x}(2x^{1} - 4) = (2x - 4)e^{x2 - 4x}

So (2x - 4)e^{x2 - 4x} is the derivative of e^{x2 - 4x}.