Which rule applies when taking the derivative of a function defined by a fraction?

When taking the derivative of a function that is defined as a fraction, the Quotient Rule is the appropriate choice. This rule specifically applies to functions that can be expressed as one function divided by another, such as ( \frac{u(x)}{v(x)} ), where both ( u ) and ( v ) are differentiable functions.

The Quotient Rule states that the derivative of a quotient of two functions is calculated as follows:

[

\frac{d}{dx} \left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}

]

Here, ( u' ) and ( v' ) represent the derivatives of ( u ) and ( v ), respectively. This formula is essential because it accounts for how changes in both the numerator ( u ) and the denominator ( v ) affect the overall fraction.

In contrast, the other rules mentioned are used for different scenarios. The Sum Rule is applied when differentiating the sum of two functions, the Chain Rule is used for composite functions, and the Product Rule is applicable when taking the derivative of the product of two functions. Each of these rules serves