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This text is written in Ukrainian and refers to the **theorem for the derivative of the quotient of functions**. The expression in parentheses \( \left( \frac{u}{v} \right)' \) represents the derivative of the quotient \( u/v \).

The rule for the derivative of a quotient is given by the **quotient rule** in calculus:

\[
\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2},
\]
where:
- \( u \) and \( v \) are differentiable functions of \( x \),
- \( u' \) is the derivative of \( u \),
- \( v' \) is the derivative of \( v \),
- \( v \neq 0 \) to ensure the denominator is non-zero.

Would you like me to explain how this rule is derived or provide an example?

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### Related Questions:
1. What is the product rule for derivatives, and how does it relate to the quotient rule?
2. How do you apply the quotient rule to trigonometric functions, such as \( \frac{\sin x}{\cos x} \)?
3. Can you demonstrate the quotient rule using an example with polynomials?
4. What happens if the denominator \( v \) equals zero in the quotient rule?
5. How does the quotient rule extend to higher-order derivatives?

**Tip:** Always check that the denominator in the quotient rule does not equal zero to avoid undefined expressions.

Теорема про похідну частки функцій \( \left( \frac{u}{v} \right)' \).

Explore the theorem for the derivative of the quotient of functions in calculus. Learn how to apply the quotient rule, given by (u/v)' = (u'v - uv') / v^2, with detailed explanations and examples. Suitable for Grades 11-12 and early university students.

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