Understanding and Applying the Quotient Rule in Calculus: Step-by-Step Guide and Examples

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. It is derived from the rules of differentiation and is particularly useful when differentiating rational functions.

The quotient rule states that if we have a function f(x) = g(x) / h(x), where g(x) and h(x) are both differentiable functions and h(x) is not equal to zero, then the derivative of f(x) can be found using the following formula:

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

To understand how this formula is derived, let’s break it down step by step:

Step 1: Start with the function f(x) = g(x) / h(x).

Step 2: Multiply the numerator and denominator of f(x) by h(x) to obtain an equivalent form of the function:

f(x) = g(x) * [h(x)]^(-1)

Step 3: Now, we can use the product rule to calculate the derivative of f(x). The product rule states that if we have a function u(x) multiplied by another function v(x), the derivative of their product is given by:

(u * v)’ = u’ * v + u * v’

Applying the product rule, we have:

f'(x) = [g(x) * [h(x)]^(-1)]’ = g'(x) * [h(x)]^(-1) + g(x) * [-1 * [h(x)]^(-2) * h'(x)]

Simplifying further, we get:

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

And that’s the quotient rule!

To use the quotient rule, follow these steps:

1. Identify the numerator and denominator functions in the quotient.
2. Differentiate the numerator function to find its derivative.
3. Differentiate the denominator function to find its derivative.
4. Use the quotient rule formula to find the derivative of the entire quotient by substituting the derivatives into the formula.

Remember to simplify your answer if possible by canceling any common factors between the numerator and denominator, or by combining like terms.

The quotient rule is a powerful tool in calculus for finding the derivative of functions that involve divisions, and it can be used to solve a variety of problems in mathematical and scientific applications.

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