Wie lautet die totale Ableitung der Funktion \(z=\ln \left(x^{4}+y\right)\) mit \(y=x^{2}\) an der Stelle \(x=1 ?\)

The 'Kettenregel' or chain rule is a fundamental concept in calculus used to differentiate compositions of functions. It's like peeling layers off an onion where you take one derivative at a time. For instance, in our exercise, we have a composite function: \[ z = \ln(x^{4} + y), \text{ where } y = x^{2} \]

Here, we need to differentiate 'z' with respect to 'x.' First, we take the derivative of 'z' with respect to the inner function 'u = x^{4} + y,' then differentiate 'u' with respect to 'x' and multiply them together. This is a direct application of the chain rule, expressed mathematically as:
\[ \frac{dz}{dx} = \frac{dz}{du} \cdot \frac{du}{dx} \]

This rule is immensely useful when dealing with nested functions and makes the differentiation process systematic and straightforward.

Differentiation is the process of finding the derivative of a function, which essentially measures how the function's value changes as its input changes. In the context of our problem, we are differentiating \[ z = \ln(x^{4} + y) \] with respect to 'x.'

To do this, we need to:

• Differentiate the outer function (natural logarithm) with respect to its argument.
• Differentiate the argument \((x^{4} + y)\) with respect to 'x.'

In the exercise, this leads us to:
\[ \frac{dz}{du} = \frac{1}{u} \quad \text{and} \quad \frac{du}{dx} = 4x^{3} + 2x \] We then combine these derivatives to get the total derivative of 'z' with respect to 'x.' This systematic process is the essence of differentiation, helping us understand how functions change and behave.

Mathematical functions are expressions that assign a unique output for every input. In this exercise, we deal with two functions: \[ z = \ln(x^{4} + y) \quad \text{and} \quad y = x^{2} \] These functions tell us how one variable depends on another. For example, 'y' is a function of 'x,' and 'z' is a function of both 'x' and 'y.' This nested relationship requires us to understand each function's role to perform the correct differentiation.

Functions can be simple like \ y = x^2 \ or more complex like \ z = \ln(x^{4} + y) \, involving compositions where one function is nested inside another. Understanding these relationships is crucial in higher mathematics and informs much of differential calculus.

A logarithmic function is the inverse of the exponential function. In our exercise, we have \[ z = \ln(x^{4} + y) \]

The natural logarithm function, denoted as \ln, is commonly used in calculus. It has unique properties that simplify differentiation, particularly because:

• Its derivative is \ \frac{d}{dx} \ln(x) = \frac{1}{x}

In our problem, recognizing this property helps us easily determine that the derivative of \ \ln(u) \ is \ \frac{1}{u} \. This understanding expedites the differentiation process, making logarithmic functions an essential tool in calculus. They often appear in problems involving rates of change and growth processes, hence their significance in many real-world applications.