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We are given: The natural exponential function is its own derivative.

We know that :

$$\begin{gathered} ddxdx=limh鈫�0ax+h-axh \\ =limh鈫�0axah-axh \\ =axlimh鈫�0ah-1h=ax.\left(cons\tan t\right) \\ =limh鈫�0ah-1h \end{gathered}$$

We see we are left with the limit h but not x, which means whatever $limh鈫�\frac{0(ah-1)}{h}$is, provided it exists, we know that it is a number or a constant so we can say: The derivative of an exponential function is a constant time itself

The number denoted by ee, called Euler's number is defined to be the number satisfying the relation :$limh鈫�0eh-1h=1$

The natural exponential function is its own derivative, in other words, we can write :

$$\begin{gathered} ddxex=ex \\ ddxex=limh鈫�0ex+h-exh \\ =limh鈫�0exeh-exh \\ =exlimh鈫�0eh-1h \\ =ex\left(cons\tan t\right) \end{gathered}$$

So, we get :$limh鈫�0eh-1h=ex$

Hence ex is its own derivative so the slope of the plot ex is the same as height or the same as its second coordinate