Product Rule Implementation and Applications

The second page delves into the produktregel (product rule) and its practical applications in differentiation. The content provides comprehensive coverage of how to handle products of functions.

Definition: For differentiable functions u and v, the product rule states that the derivative of their product f(x) = u(x)·v(x) is given by f'(x) = u'(x)·v(x) + u(x)·v'(x).

Example: For h(x) = x²·cos(x):

• u(x) = x² with u'(x) = 2x
• v(x) = cos(x) with v'(x) = -sin(x)
• h'(x) = 2x·cos(x) - x²·sin(x)

Highlight: The product rule is derived from the fundamental definition of derivatives and is essential for differentiating products of functions.

Vocabulary:

• Ableitung: Derivative
• Diffbar: Differentiable
• Produktregel: Product rule

Chain Rule Fundamentals and Applications

This page introduces the fundamental concepts of the kettenregel (chain rule) and its practical applications in calculus. The content explores how to differentiate composite functions using various approaches.

Definition: The chain rule states that for a composite function f(x) = g(h(x)), the derivative is f'(x) = g'(h(x)) · h'(x), where g and h are differentiable functions.

Example: For f(x) = (3x + 5)⁴, using the chain rule:

• g(x) = x⁴ (outer function)
• h(x) = 3x + 5 (inner function)
• f'(x) = 4(3x + 5)³ · 3 = 12(3x + 5)³

Highlight: Two distinct approaches are presented:

1. Direct application of chain rule (faster and simpler)
2. Expansion followed by differentiation (useful for verification)

Vocabulary:

• Äußere Funktion: Outer function
• Innere Funktion: Inner function
• Diffbar: Differentiable