Question

Prove that if the functions \(f_{i}: E^{1} \rightarrow E^{*}(C)\) are differentiable at \(p,\) so is their product, and $$ \left(f_{1} f_{2} \cdots f_{m}\right)^{\prime}=\sum_{i=1}^{m}\left(f_{1} f_{2} \cdots f_{i-1} f_{i}^{\prime} f_{i+1} \cdots f_{m}\right) \text { at } p $$

Short answer

The derivative at \(p\) is \(\sum_{i=1}^{m} f_1(p) \cdots f_{i-1}(p) f_i'(p) f_{i+1}(p) \cdots f_m(p)\).

Step-by-step solution

Understand the Product Rule for Differentiation

The product rule for differentiation states that the derivative of a product of functions is given by applying differentiation to each factor while keeping other factors constant and then summing up these derivatives. For two functions \(g(x)\) and \(h(x)\), the derivative \((g\cdot h)'(x)\) is \(g'(x)h(x) + g(x)h'(x)\). We will extend this to multiple functions.

Express the Product of Functions

Consider the product of \(m\) differentiable functions \(f_{1}, f_{2}, \, \ldots, \, f_{m}\). The function can be expressed as a single function \(F(x) = f_1(x)f_2(x) \cdots f_m(x)\). Our task is to find the derivative of this function at a point \(p\).

Apply the Product Rule Iteratively

To differentiate the product of multiple functions, apply the product rule iteratively: Fix all functions except one in their derivative form. The derivative of \(F\) at \(p\) is given by:\[ F'(p) = \sum_{i=1}^{m} f_1(p) \cdots f_{i-1}(p) f_i'(p) f_{i+1}(p) \cdots f_m(p) \] This expression means you obtain the derivative by differentiating each \(f_i\) at \(p\), multiplying it by the remaining non-differentiated functions.

Perform the Differentiation

Differentiate each \(f_i\) at \(p\), multiply it by the remaining functions (excluding \(f_i\)), and sum the results:\[ (f_1 f_2 \cdots f_m)'(p) = f_1(p)f_2(p)\cdots f_{i-1}(p)f_i'(p)f_{i+1}(p)\cdots f_m(p) \] Repeat this for each function \(i\) in the product to account for all parts in the sum. This results in the given formula.

Verify the Formula

Verify that the formula correctly represents the derivative of a product of functions. This is done by ensuring that every function has been differentiated exactly once and all such terms are summed up, which corresponds to the complete application of the product rule for each \(f_i\).

Key concepts

Differentiable Functions

In calculus, a function is considered differentiable at a point if it has a derivative at that point. This means that the function has a well-defined tangent at that exact location and exhibits a smooth, continuous change. Differentiability is a property that ensures functions behave predictably, allowing for accurate calculations of rates of change, among other dynamics.
A function is more than just its ability to produce numbers from inputs; it reflects how these inputs change with respect to one another. Therefore, establishing differentiability provides a foundation for performing operations like differentiation, crucial in mathematical analysis and applications.

• The function must be continuous at the point under consideration.
• It should not have any sharp edges or corners at that point.

Understanding differentiability is critical when dealing with the derivatives of multiple functions, as this concept forms the basis for applying the product rule effectively.

Elias Zakon

Elias Zakon was a respected mathematician whose contributions clarified many aspects of mathematical logic and analysis. His work often focused on the underlying principles that govern mathematical expressions and functions. Zakon's research laid the groundwork for understanding complex mathematical concepts, making significant advances in how functions and derivatives are interpreted today.
His approach to teaching mathematics was revolutionary, emphasizing clarity and intuition, which is reflected in how many modern mathematical theorists approach problems.
Zakon's influence is still felt when students learn about product rules and differentiation, as his principles guide logical structuring of mathematical proofs and problem solving.

• Promoted logical analysis in mathematical theory.
• Influenced modern teaching methods in mathematics.
• Clarified foundational principles of calculus and function theory.

His insights help students not just to perform calculations, but also to understand the meaning behind them, especially in derivative structures.

Derivative of Multiple Functions

When dealing with the derivative of multiple functions, we are referring to finding the rate at which a product of several differentiable functions changes with respect to one variable. This brings us to the product rule, a fundamental technique in calculus.
The product rule provides a way to differentiate a product of two or more functions. For instance, if we have functions like \( f(x), g(x), h(x) \), the derivative of their product is not simply the product of their derivatives. Instead, you apply the rule by taking turns differentiating each function while keeping the others constant and then summing up the results.

• Ensures all factors in the product have been accounted for in the derivative.
• Facilitates complex calculations by reducing errors in differentiation.
• Extends to numerous functions through iterative application.

This requirement to iterate through each function demands precision, ensuring all derivatives are accounted for in multi-function products.

Mathematical Proof

In mathematics, a proof is a logical argument that verifies the truth of a statement. Proving the formula for the derivative of a product of functions involves using established rules systematically, like the product rule.
Depending on the complexity, proofs can vary in length, but each step should follow logically from the last, ensuring clarity and correctness. When proving the derivative of multiple functions, the challenge lies in expressing and applying the product rule iteratively. This requires a clear understanding of both the logical flow and the mathematical techniques involved.
Mathematical proofs are vital since they affirmatively test the validity of mathematical principles, confirming that rules and formulas used in problem-solving are reliable.

• Provides assurance of correctness in mathematical computations.
• Builds a framework of logic used in further mathematical explorations.
• Requires a comprehensive understanding of laws and their application.

For students, understanding how to construct and follow a proof fosters deeper comprehension and promotes critical thinking in mathematical contexts.