$$ z=f(x ; y)=5 x \cdot e^{-x y}+\ln \sqrt{x^{2}+y^{2}}+\cos (\pi x+y) $$ Berechnen Sie: \(z_{x}(1 ; 0), z_{y}(0 ; 1), z_{x y}(-1 ; 0), z_{y y}(5 ; 0), z_{x y x}(-1 ; 0)\)

The product rule is essential when differentiating functions formed by the multiplication of two or more terms. It states that if you have two functions, say \(u(x)\) and \(v(x)\), their derivative is:
\[ (uv)' = u'v + uv' \]
In our exercise, we encountered the term \(5x e^{-xy}\). Here, \(u(x) = 5x\) and \(v(x) = e^{-xy}\). Applying the product rule, we differentiate each part as follows:

• \(u'(x) = 5\)
• \(v'(x) = -y e^{-xy}\)

Thus, the combined derivative is:
\[ \frac{d}{dx} [5x e^{-xy}] = 5 e^{-xy} + 5x \times (-y e^{-xy}) = 5 e^{-xy} - 5xy e^{-xy} \]
This same principle is applied throughout the function to ensure accurate differentiation.

Logarithmic differentiation simplifies the differentiation of more complex functions, especially those involving logarithms. It leverages the properties of logarithms to break down complex expressions into more manageable parts. In our case, we used the term \( \ln \sqrt{x^2 + y^2} \).
First, let's simplify \( \ln \sqrt{x^2 + y^2} \) to \( \frac{1}{2} \ln (x^2 + y^2) \). Now, we differentiate with respect to \( x \):

• The chain rule gives us the differentiation part
• We get \( \frac{1}{2} \times \frac{2x}{x^2 + y^2} = \frac{x}{x^2 + y^2} \)

This simplification streamlines the process, making it easier to understand and apply. Logarithmic differentiation is powerful for handling expressions that are products, quotients, or powers of functions.

Understanding higher-order derivatives is crucial for analyzing the behavior of functions beyond their first derivatives. Higher-order derivatives are simply derivatives of derivatives. For our exercise, we computed various second and third-order partial derivatives, such as \(z_{xy}\) and \(z_{xyx}\).

• Found by differentiating the first derivative, \(z_x\), with respect to another variable \( y \)
• This sequential differentiation helps us understand how the function changes in a more granular way

For instance, to find \( z_{xy} \), we differentiated \( z_x \) with respect to \( y \):
Given that \( z_x = 5e^{-xy} - 5xye^{-xy} + \frac{x}{x^2 + y^2} - \pi sin(\pi x + y) \), we partially differentiate with respect to \( y \), yielding:
\( z_{xy} = -5xe^{-xy} + 5xye^{-xy} - \frac{2xy}{(x^2 + y^2)^2} - \pi cos(\pi x + y) \).

This hierarchical process continues the analysis, exploring the deeper changes in the function and giving more insight into its behavior.