## Quotient Rule

### The quotient rule is a formula used to find the derivative of a quotient of two functions

The quotient rule is a formula used to find the derivative of a quotient of two functions. Let’s consider two functions: f(x) and g(x). The quotient rule states that the derivative of their quotient, (f(x) / g(x)), can be obtained by the following formula:

(f'(x) * g(x) – f(x) * g'(x)) / [g(x)]^2

Here, f'(x) represents the derivative of f(x), and g'(x) represents the derivative of g(x).

To use the quotient rule, you need to follow these steps:

1. Identify the functions f(x) and g(x).

2. Take the derivative of f(x) and g(x) individually.

3. Apply the quotient rule formula to find the derivative of (f(x) / g(x)), using the derivatives you obtained in step 2.

4. Simplify the resulting expression, if possible.

To better understand the quotient rule, let’s go through an example:

Example:

Consider the function y(x) = (3x^2 + 2x + 1) / (2x^2 – 5x + 3).

To find the derivative of this function using the quotient rule, we first need to identify f(x) and g(x):

f(x) = 3x^2 + 2x + 1

g(x) = 2x^2 – 5x + 3

Next, we find their derivatives:

f'(x) = 6x + 2

g'(x) = 4x – 5

Now, we can use the quotient rule formula to find the derivative of y(x):

y'(x) = [(f'(x) * g(x)) – (f(x) * g'(x))] / [g(x)]^2

= [(6x + 2) * (2x^2 – 5x + 3) – (3x^2 + 2x + 1) * (4x – 5)] / [(2x^2 – 5x + 3)]^2

Simplifying this expression further might not be necessary unless explicitly requested. However, if simplification is desired, one can expand and combine like terms in the numerator, and simplify the denominator using algebraic techniques.

That’s the essence of the quotient rule. It allows us to find the derivative of a quotient of functions without having to use the limit definition of derivative.

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