Maximizing Derivatives With The Quotient Rule – A Comprehensive Guide

Quotient Rule

d/dx [f(x)/g(x)] = [g(x)f'(x) – g'(x)f(x)]/[g(x)²]

The quotient rule is a formula for finding the derivative of a function that represents the ratio of two other functions. Let’s say we have two functions, f(x) and g(x), where g(x) ≠ 0, and we want to find the derivative of their ratio, h(x) = f(x)/g(x).

The general formula for the quotient rule is:

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

In other words, to find the derivative of h(x), we need to take the following steps:

1. Multiply the denominator function (g(x)) squared.
2. Multiply the numerator function f(x) by the derivative of the denominator function g'(x).
3. Multiply the denominator g(x) by the derivative of the numerator f'(x).
4. Subtract the results of step 2 from step 3.
5. Divide the result of step 4 by the denominator function squared from step 1.

Here is an example to help illustrate the quotient rule:

Let’s say we have two functions f(x) = x^2 and g(x) = x + 1, and we want to find the derivative of their ratio h(x) = x^2 / (x + 1).

First, we need to identify f'(x) and g'(x):
f'(x) = 2x (by using the power rule)
g'(x) = 1 (by using the derivative of a sum rule)

Then, we can apply the quotient rule:

h'(x) = [(x + 1) * 2x – x^2 * 1] / (x + 1)^2
h'(x) = (2x^2 + 2x – x^2) / (x + 1)^2
h'(x) = (x^2 + 2x) / (x + 1)^2

Therefore, the derivative of h(x) = x^2 / (x + 1) is h'(x) = (x^2 + 2x) / (x + 1)^2.

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