For a special case, y = e x, its derivative can be extended to a more general exponential function.

According to the definition of derivative, give the independent variable x a tiny increment dx, and you can get:

Expand the above formula, and then bring up E x, and you get:

Observing the above formula, we will find that the pile on the right of e x is formula (1) (where dx tends to 0), and the value of formula (1) is 1, so the derivative of y = e x is itself, e x.

After solving this special example, let's look at the more general exponential function y = a x (a is an arbitrary real number).

Here we need a trick to write A as E lnA (where LN is the natural logarithm with E as the base), so there are:

It is easy to see that this is a composite function. According to the chain derivative law, we can get:

Don't forget, LNA. Therefore, given an arbitrary exponential function y = a x, its derivative is (a x) ln a

Extended data

Basic derivative formula

Give the increment of the independent variable

Get function increment

upper left

Seek the limit

Four Derivative Algorithms and Their Properties

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