What is the quotient rule for derivatives?

The quotient rule is a fundamental formula in calculus used to find the derivative of a function that is in the form of a quotient of two differentiable functions. It's a crucial tool for differentiating composite functions and is often used in problems involving rates of change and optimization.

Given two differentiable functions, $u(x)$ and $v(x)$, the quotient rule states that the derivative of the quotient $\frac{u(x)}{v(x)}$ is given by:

$\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \frac{du}{dx} - u(x) \frac{dv}{dx}}{v^2(x)}$

In other words, to find the derivative of a quotient, you multiply the denominator by the derivative of the numerator, subtract the numerator multiplied by the derivative of the denominator, and then divide by the square of the denominator.

Here's a step-by-step breakdown:

1. Identify $u(x)$ and $v(x)$ in the given function.
2. Find the derivatives $\frac{du}{dx}$ and $\frac{dv}{dx}$.
3. Multiply the derivative of the numerator by the denominator: $v(x) \frac{du}{dx}$.
4. Multiply the numerator by the derivative of the denominator: $u(x) \frac{dv}{dx}$.
5. Subtract the result of step 4 from the result of step 3: $v(x) \frac{du}{dx} - u(x) \frac{dv}{dx}$.
6. Divide the result by the square of the denominator: $\frac{v(x) \frac{du}{dx} - u(x) \frac{dv}{dx}}{v^2(x)}$.