What is e exponential

What are e-functions? Deriving and antiderivatives easily explained

The e-function: properties

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) $

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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:

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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