The Quotient Rule: A Comprehensive Guide to Calculus Differentiation of Quotient Functions

Quotient Rule

The quotient rule is a formula used to differentiate functions that are in the form of a quotient or division of two functions

The quotient rule is a formula used to differentiate functions that are in the form of a quotient or division of two functions. It is one of the most important rules in calculus.

The quotient rule states that if you have a function in the form:

f(x) = g(x) / h(x)

where g(x) and h(x) are differentiable functions, then the derivative of f(x) can be found using the following formula:

f'(x) = (h(x) * g'(x) – g(x) * h'(x)) / (h(x))^2

Let’s break down the components of the formula:

– g(x) is the numerator function
– h(x) is the denominator function
– g'(x) is the derivative of the numerator function with respect to x
– h'(x) is the derivative of the denominator function with respect to x

To find the derivative of f(x), we need to follow these steps:

1. Differentiate g(x) to find g'(x).
2. Differentiate h(x) to find h'(x).
3. Apply the quotient rule formula to find f'(x).

Here is an example to illustrate the application of the quotient rule:

Let’s say we have the function:

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

To find the derivative, we follow these steps:

1. Differentiate the numerator, g(x) = x^2:

g'(x) = 2x

2. Differentiate the denominator, h(x) = 2x + 3:

h'(x) = 2

3. Apply the quotient rule formula:

f'(x) = (h(x) * g'(x) – g(x) * h'(x)) / (h(x))^2

Substituting the values:

f'(x) = ((2x + 3) * 2x – x^2 * 2) / (2x + 3)^2

Simplifying further:

f'(x) = (4x^2 + 6x – 2x^2) / (2x + 3)^2
= (2x^2 + 6x) / (2x + 3)^2

So, the derivative of f(x) = x^2 / (2x + 3) is f'(x) = (2x^2 + 6x) / (2x + 3)^2

Remember, the quotient rule is an essential tool for finding the derivative of functions that involve division. Practice using this rule with various functions to build your understanding and mastery of calculus concepts.

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