Math Problem Statement

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

Solution

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

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

(uv)=uvuvv2,\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}, where:

  • uu and vv are differentiable functions of xx,
  • uu' is the derivative of uu,
  • vv' is the derivative of vv,
  • v0v \neq 0 to ensure the denominator is non-zero.

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


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 sinxcosx\frac{\sin x}{\cos x}?
  3. Can you demonstrate the quotient rule using an example with polynomials?
  4. What happens if the denominator vv 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.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Quotient Rule

Formulas

(u/v)' = (u'v - uv') / v^2

Theorems

Quotient Rule for Derivatives

Suitable Grade Level

Grades 11-12 / Early University