# Quotient Rule

## The Simplified Guide to Understanding the Quotient Rule

The quotient rule is an essential tool for differentiating functions that are expressed as one function divided by another. In order to better understand this formula, let's take a look at its structure and practice using it through some examples.

### Understanding Quotient Functions

A quotient function is a type of mathematical expression where one function is divided by another. This can be written as f(x) = g(x)/h(x). In order to differentiate this type of function, the quotient rule comes into play.

### The Formula for the Quotient Rule

The equation for the quotient rule is as follows:

**dy/dx = (v du/dx - u dv/dx)/v^2 if y = u/v**

In function notation, it can also be written as:

**If f(x) = g(x)/h(x), then f'(x) = (h(x)g'(x) - g(x)h'(x))/(h(x))^2**

### Examples of Using the Quotient Rule

Let's take a look at some problems and see how the quotient rule can be applied.

**Problem 1:**If y = (x^2 + 2x)/3x, find dy/dx.

To start, we can break down the equation and identify the necessary values for the quotient rule:

- v = 3xdu/dx = 2x + 2dv/dx = 3

**Problem 2:**If y = sin(x)/x, find dy/dx.

Since the function is written in function notation and involves a fraction, it is evident that the quotient rule formula in function notation is needed:

- v = xdu/dx = cos(x)dv/dx = 1

By substituting these values into the formula, we can solve for dy/dx.

### Using Function Notation in the Quotient Rule

It's important to be able to use the quotient rule in function notation, as it may appear in exam questions. Let's review the formula in terms of function notation:

**If f(x) = g(x)/h(x), then f'(x) = (h(x)g'(x) - g(x)h'(x))/(h(x))^2**

### Solving Problems through the Quotient Rule

Let's apply the quotient rule to solve a problem and find a specific value.

**Problem: Find the value of f'(2) for the point (2, 1/3) on the curve where f(x) = (x^2 + 2x)/3x.**

First, we can identify the necessary elements of the formula and calculate them:

**v = 3xdu/dx = 2x + 2dv/dx = 3**

Next, we can substitute these values into the formula to solve for dy/dx.

**dy/dx = ((3x)(2x + 2) - (x^2 + 2x)(3))/(3x)^2**

Finally, we can substitute x = 2, since that is the given x-coordinate, to find f'(2).

**dy/dx = ((3(2))(4 + 2) - (4 + 4)(3))/(3(2))^2= (18 - 24)/36= -1/6**

Therefore, f'(2) = -1/6.

### Key Takeaways

- The quotient rule is used for differentiating quotient functions.
- A quotient function is a function expressed as one divided by another.
- The formula for the quotient rule is dy/dx = (v du/dx - u dv/dx)/v^2 if y = u/v
- This can also be written in function notation: If f(x) = g(x)/h(x), then f'(x) = (h(x)g'(x) - g(x)h'(x))/(h(x))^2