Bernoulli's formula is a key concept in mathematics used to express the n-th derivative of the product of two functions. It is particularly useful in integration and differential equations. The formula is expressed as:

Bernoulli's Formula:

$\frac{d^n}{dx^n}(u \cdot v) = \sum_{k=0}^{n} \binom{n}{k} \cdot \frac{d^{n-k}u}{dx^{n-k}} \cdot \frac{d^k v}{dx^k}$

Where:

• $u$ and $v$ are functions of $x$,
• $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$,
• $\frac{d^m u}{dx^m}$ and $\frac{d^m v}{dx^m}$ denote the m-th derivatives of $u$ and $v$, respectively.

This formula arises in the generalization of the product rule for higher-order derivatives. It shows how the n-th derivative of a product involves all combinations of derivatives of $u$ and $v$ up to order $n$.

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Related Questions:

1. What is the binomial coefficient, and how is it calculated?
2. How does Bernoulli's formula apply in physics or engineering problems?
3. Can you demonstrate Bernoulli's formula with a specific example (e.g., $u = x^2, v = e^x$)?
4. How is Bernoulli's formula derived from the product rule?
5. What are common mistakes made when applying Bernoulli's formula?

Tip: For clarity, always double-check the indices of differentiation to ensure no steps are missed in applying Bernoulli's formula.