# What is e exponential

## What are e-functions? Deriving and antiderivatives easily explained

### monotony

The exponential function is strictly monotonically growing and the growth is exponential. This means that the function increases very quickly. The larger \$ x \$, the larger the \$ y \$ value, as we can see on the figure:

Figure: e-function, rapid growth

### Points of intersection with the axes

The exponential function has no zeros because a power can never be zero. So \$ f (x) \$ = \$ e \$ always applies x ≠ \$ 0 \$. As \$ x \$ becomes smaller, your graph approaches the \$ x \$ axis more and more and \$ \ lim \ limits_ {x \ to -∞} \$ \$ e \$ appliesx = \$ 0 \$. So this axis is an even asymptote.

The graph of this function intersects the \$ y \$ -axis at position 1, since \$ f (0) \$ = \$ e \$0 = \$ 1 \$ is.

### Inverse function

The inverse function of the exponential function is the natural logarithm function. \$ f (x) = e ^ x \$, \$ f ^ {- 1} (x) = ln (x) \$

Inverse function of \$ f (x) = e ^ x \$
\$ f ^ {- 1} (x) = \ log_e (x) = ln (x) \$

Figure: Functions \$ \ rightarrow f ^ {- 1} (x) = ln (x) \$. Both are inverse functions and thus mirror images of each other on the straight line \$ y \$ = \$ x \$.

### Set of definitions and values

We can substitute any real number for \$ x \$. That means thatDefinition set is: \$ D_f = \ mathbb {R} \$

As we can see from the graph, it runs above the x axis, which is asymptote. The range of values ​​is therefore: \$ W_f = \ mathbb {R ^ +} \$. These are all positive real numbers.

### Derive the exponential function and form an antiderivative

The derivation and also the antiderivative of the exponential function form an exponential function again:

Derivation: \$ f '(x) = e ^ x \$
Antiderivative: \$ F (x) = e ^ x \$

But why is this the case with the e-function?

The general derivation of exponential functions is: \$ f (x) = a ^ x \$ \$ \ rightarrow f '(x) = a ^ x \ cdot ln (a) \$

Applying this to \$ f (x) = e ^ x \$ we get:

\$ f '(x) = (e ^ x)' = e ^ x \ cdot ln (e) = e ^ x \ cdot 1 = e ^ x \$

With theExercises you can test your newly acquired knowledge of deriving exponential functions. I wish you good luck!

Video: Simon Wirth

Text: Chantal Rölle