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.