Die Funktionen f . g: G ⊂𝐑^' →𝐑(G offen) mögen in ξ∈G Gradienten besitzen. also nach allen Verän

step 1 In the question we are given that f -dice x is greater than 0 so this implies f x is a increasin

Die Funktionen f . g: G ⊂𝐑^' →𝐑(G offen) mögen in ξ∈G...

Die Funktionen $f . g: G \subset \mathbf{R}^{\prime} \rightarrow \mathbf{R}(G$ offen) mögen in $\xi \in G$ Gradienten besitzen. also nach allen Veränderlichen partiell differenzierbar sein. Dann trifft dies auch für $f+g, \alpha f . f g$ und. falls $g(\xi) \neq 0$ ist, für $f / g \mathrm{zu}$, und es gelten die folgenden Formeln. bei denen als Argument überall $\xi$ einzutragen ist: $$ \begin{aligned} &\operatorname{grad}(f+g)=\operatorname{grad} f+\operatorname{grad} g, \quad \operatorname{grad}(\alpha f)=\alpha \operatorname{grad} f \\ &\operatorname{grad}(f g)=f \operatorname{grad} g+\operatorname{ggrad} f . \quad \operatorname{grad} \frac{f}{g} \quad=\frac{g \operatorname{grad} f-f \operatorname{grad} g}{g^{2}} \end{aligned} $$

Die Funktionen $f . g: G \subset \mathbf{R}^{\prime} \rightarrow \mathbf{R}(G$ offen) mögen in $\xi \in G$ Gradienten besitzen. also nach allen Veränderlichen partiell differenzierbar sein. Dann trifft dies auch für $f+g, \alpha f . f g$ und. falls $g(\xi) \neq 0$ ist, für $f / g \mathrm{zu}$, und es gelten die folgenden Formeln. bei denen als Argument überall $\xi$ einzutragen ist:
$$
\begin{aligned}
&\operatorname{grad}(f+g)=\operatorname{grad} f+\operatorname{grad} g, \quad \operatorname{grad}(\alpha f)=\alpha \operatorname{grad} f \\
&\operatorname{grad}(f g)=f \operatorname{grad} g+\operatorname{ggrad} f . \quad \operatorname{grad} \frac{f}{g} \quad=\frac{g \operatorname{grad} f-f \operatorname{grad} g}{g^{2}}
\end{aligned}
$$

Key Concepts

Gradient

The gradient of a scalar function is a vector that contains all of its partial derivatives. It represents both the direction and rate of the fastest increase of the function and is a fundamental tool in multivariable calculus.

Partial Differentiability

A function is partially differentiable if, with respect to each variable individually, the partial derivative exists at a given point. This concept is essential because the existence of all partial derivatives in a neighborhood is a requirement for a function to have a gradient.

Algebraic Operations on Differentiable Functions

When functions are partially differentiable, operations such as addition, scalar multiplication, product, and division (when the denominator is non-zero) preserve differentiability. These properties allow the gradient of combined functions to be computed using specific formulas.

Product Rule

The product rule provides the method to compute the gradient of the product of two functions. It states that the gradient of the product is equal to one function multiplied by the gradient of the other, plus the other function multiplied by the gradient of the first. This rule is crucial in applications involving the multiplication of functions.

Quotient Rule

The quotient rule describes how to differentiate a quotient of two functions that are both differentiable, with the denominator not equal to zero. It expresses the gradient of the quotient in terms of the gradients of the numerator and denominator, accounting for the square of the denominator, and is essential in handling ratios of functions.

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