**Derivative of exponential functions**

**Euler's number**

is one of the central irrational constants in mathematics, like π . Euler’s number has the following important feature. When set as the base number of an exponential function, the derivative of the function is itself.

**f(x)**

**f(x)****f'(x)**

**f'(x)**The e-base logarithm is called the natural logarithm. It is denoted as

**Example 1**

Example 2

In this case, the exponent has a function *2x*, so when we differentiate, remember to multiply by the derivative of the inner function.

**Example 3**

Differentiate

We use the Euler's number and the natural logarithm and denote

Now we can differentiate

In general, the derivative of an exponential function is

**Example 4**

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