Q. 4 - Calculus

Question

The natural exponential function is its own derivative. Explain what this means graphically. (Use words like “height” and “slope.”)

Step-by-Step Solution

Answer

We can say that the natural exponential function is its own derivative means the slope of the plot $e^x$ is the same as its height or the same as its second coordinate.

Step 1. Given Information

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

Step 2. Definition of derivative.

We know that :

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

We see we are left with the limit h but not x, which means whatever $limh\to \frac{0(ah-1)}{h}$ 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 : $limh\to 0eh-1h=1$

Step 3 . Graphical meaning

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

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

So, we get :$limh\to 0eh-1h=ex$

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