# What is the differentiation of x

## Differentiate or derive a function

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The slope of a function at a point \ (x \) can be calculated using the differential quotient. One calls this calculation Deriving a function or Differentiate.
The derivative of a differentiable function at the point \ (x_0 \) is given by \ [f '(x_0) = \ lim _ {\ Delta x \ rightarrow 0} \ frac {f (x_0 + \ Delta x) -f (x_0)} {\ Delta x} \] defined.
Determining the derivative of a function with the differential quotient is often very difficult. Therefore, formulas, the so-called derivation rules, are usually used for such calculations.
Note: The differential quotient can be calculated for a large number of functions, but not for all.

### Examples of derivatives with the differential quotient

What is the slope of the function \ (f (x) = x ^ 2 \) at the point \ (x_0 = 1 \)?
The differential quotient \ (f '(x_0) \) indicates the slope at the point \ (x_0 \). \ [f '(x_0) = \ lim _ {\ Delta x \ rightarrow 0} \ frac {f (x_0 + \ Delta x) -f (x_0)} {\ Delta x} \] \ [f' (x_0) = \ lim _ {\ Delta x \ rightarrow 0} \ frac {(x_0 + \ Delta x) ^ 2-x_0 ^ 2} {\ Delta x} \] \ [f '(x_0) = \ lim _ {\ Delta x \ rightarrow 0} \ frac {(x_0 ^ 2 + 2 \ cdot x_0 \ Delta x + (\ Delta x) ^ 2) -x_0 ^ 2} {\ Delta x} \] \ [f '(x_0) = \ lim _ { \ Delta x \ rightarrow 0} \ frac {2 \ cdot x_0 \ Delta x + (\ Delta x) ^ 2} {\ Delta x} \] \ [f '(x_0) = \ lim _ {\ Delta x \ rightarrow 0 } (2 \ cdot x_0 + \ Delta x) \] Since there is now no indefinite expression like Zero through zero results, one can set \ (\ Delta x = 0 \). \ [f '(x_0) = 2 \ cdot x_0 + 0 = 2 \ cdot x_0 \] By inserting \ (x_0 = 1 \) you get the slope at this point. The slope there is \ (f '(1) = 2 \).
What is the slope of the linear function \ (f (x) = m \ cdot x + t \) at the point \ (x_0 = -10 \)?
The differential quotient \ (f '(x_0) \) indicates the slope at the point \ (x_0 \). \ [f '(x_0) = \ lim _ {\ Delta x \ rightarrow 0} \ frac {f (x_0 + \ Delta x) -f (x_0)} {\ Delta x} \] \ [f' (x_0) = \ lim _ {\ Delta x \ rightarrow 0} \ frac {m \ cdot (x_0 + \ Delta x) + t- (m \ cdot x_0 + t)} {\ Delta x} \] \ [f '(x_0) = \ lim _ {\ Delta x \ rightarrow 0} \ frac {m \ cdot x_0 + k \ cdot \ Delta xm \ cdot x_0} {\ Delta x} \] \ [f '(x_0) = \ lim _ {\ Delta x \ rightarrow 0} \ frac {m \ cdot \ Delta x} {\ Delta x} \] \ [f '(x_0) = \ lim _ {\ Delta x \ rightarrow 0} m \] Now you can \ (\ Set delta x = 0 \). \ [f '(x_0) = m \] At the point \ (x_0 = -10 \) the slope is \ (f' (- 10) = m \).

### Derivative function

One can look at the derivative of a function as a function. This function is called the derivative function. The place where you want to know the derivative / slope is then the function variable. There are many different ways of writing the derivative function, such as the following: \ [f '(x) = \ frac {dy} {dx} = \ frac {df} {dx} = \ frac {d} {dx} f (x ) \]
It is given \ (f (x) = a \ cdot x ^ 2 \). The derivative function is \ (f '(x) = 2ax \). The function graph of the function \ (f \) and the function graph of the associated derivative function \ (f '\) are shown in the following graphic, where the parameter \ (a \) can be varied / changed with the slider.

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