Applying the Quotient Rule: Step-by-Step Example and Formula Explanation

derivative of quotient

To find the derivative of a quotient, we can use the quotient rule

To find the derivative of a quotient, we can use the quotient rule. The quotient rule is a formula used to differentiate a function that is the division of two other functions.

Let’s say we have a function f(x) = g(x) / h(x), where g(x) and h(x) are both differentiable functions.

The quotient rule states that the derivative of f(x) is given by the following formula:

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

In other words, the numerator of the derivative is the difference between the product of the derivative of the numerator function (g'(x)) and the denominator function (h(x)), and the product of the numerator function (g(x)) and the derivative of the denominator function (h'(x)). The denominator of the derivative is [h(x)]^2.

Here’s a step-by-step example to show how to apply the quotient rule:

Let’s find the derivative of f(x) = (3x^2 + 2x) / x

Step 1: Identify the numerator and denominator functions:
The numerator function g(x) = 3x^2 + 2x
The denominator function h(x) = x

Step 2: Differentiate the numerator and denominator functions:
g'(x) = 6x + 2
h'(x) = 1

Step 3: Apply the quotient rule formula:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / [h(x)]^2
f'(x) = ((6x + 2) * x – (3x^2 + 2x) * 1) / x^2
f'(x) = (6x^2 + 2x – 3x^2 – 2x) / x^2
f'(x) = (3x^2) / x^2
f'(x) = 3

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

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