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.

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 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**

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.

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**

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.

- 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

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