# 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 $x=1$:

$f'(x)=2x$

$f'(1)=2\cdot1=2$