Derivative
The Derivative function (abbreviated derivative) assigns each $x$ to the corresponding differential quotient.
Calculating the derivative is called differentiating or deriving.
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Remmeber
The derivative $f'(x)$ assigns the slope of the parent function $f$ at any point $x$.
The advantage of the derivative lies within not having to calculate the differential quotient over and over. Instead, you have a function in which you insert the point with the searched slope.
Higher derivatives
When differentiating the derivative, the derivative of the derivative function is called second derivative.
The derivative of the second derivative $f''$ is called third derivative etc.
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Hint
In general, the first three derivatives are normally written with dashes:
$f'(x)$, $f''(x)$ und $f'''(x)$
After the third derivative, the style is usually the following:
$f^{(4)}(x)$
$f^{(5)}(x)$
...
$f'(x)$, $f''(x)$ und $f'''(x)$
After the third derivative, the style is usually the following:
$f^{(4)}(x)$
$f^{(5)}(x)$
...
Example
The function $f(x)$ and its first two derivatives:
Calculating the derivative at point $x=1$:
$f'(x)=2x$
$f'(1)=2\cdot1=2$