# Why is #d/dxe^x=e^x#?

##### 2 Answers

This follows from the definition of natural logarythm and its inverse.

#### Explanation:

The "why" depends on how you've defined

#### Explanation:

**Define #lnx# first**

One approach is to define

then to define

finally, define

In this case

Differentiating implicitly gets us

So,

**Define #e^x# independently of #lnx#**

**Definition 1**

For positive

(We owe you a proof that this is well-defined.)

Then, using the definition of derivative:

We then define

With this definition we get

# = e^xlim_(hrarr0)((e^h-1)/h) = e^x#

**Definition 2**

(

(We owe you a proof that this is well defined.)

Differentiating term by term (we owe you a proof that this is possible), we get

Which simplifies to