Calculating the Derivative Using the h-Method

This page outlines the step-by-step process for finding the derivative of a function at a specific point using the h-Method, also known as the Differenzenquotient Formel. The method is demonstrated through a detailed example, providing a clear Differentialquotient Beispiel mit Lösung.

The process involves three main steps:

1. Setting up the difference quotient
2. Simplifying the expression
3. Taking the limit as h approaches zero

Definition: The Differenzenquotient is the average rate of change of a function over a small interval, represented by h.

Example: For the function f$x$ = 2x², the derivative at x = 1 is calculated using the h-Method.

The calculation begins by setting up the difference quotient:

$f(x₀ + h) - f(x₀)$ / h = $f(1 + h) - f(1)$ / h

This is then expanded and simplified:

$2(1 + h)² - 2·1²$ / h = $2(1² + 2h + h²) - 2$ / h = $2 + 4h + 2h² - 2$ / h = $4h + 2h²$ / h = 4 + 2h

Highlight: The key to the h-Method is to simplify the expression so that h can be factored out of the numerator, allowing for cancellation with the h in the denominator.

Finally, the limit is taken as h approaches zero:

lim[h→0] $4 + 2h$ = 4

Vocabulary: The Differentialquotient is the limit of the difference quotient as h approaches zero, representing the instantaneous rate of change.

This example demonstrates how the h-Method can be used to find the derivative of a function at a specific point, providing a practical application of the Differenzenquotient Differentialquotient relationship.