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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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