Mastering the Quotient Rule for Derivatives: A Step-by-Step Guide With Examples

derivative of quotient

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

To find the derivative of a quotient, you can use the quotient rule. The quotient rule states that if you have a function with the form f(x) = g(x) / h(x), then the derivative of f(x) is given by:

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

Here’s a step-by-step guide on how to apply the quotient rule:

1. Let’s say you have a function f(x) = g(x) / h(x), and you want to find its derivative.

2. Differentiate the numerator (g(x)) and the denominator (h(x)) separately.

3. The derivative of the numerator, g'(x), represents the rate at which the numerator changes with respect to x.

4. The derivative of the denominator, h'(x), represents the rate at which the denominator changes with respect to x.

5. Apply the quotient rule formula by subtracting the product of g(x) and h'(x) from the product of g'(x) and h(x).

6. Finally, divide the result by (h(x))^2 to get the derivative of the quotient.

Let’s take an example to illustrate the process:

Consider the function f(x) = (2x + 1) / (x^2 + 3x + 2).

1. Identify the numerator g(x) = 2x + 1 and the denominator h(x) = x^2 + 3x + 2.

2. Differentiate the numerator and denominator:
g'(x) = 2 (the derivative of 2x is 2)
h'(x) = 2x + 3 (the derivative of x^2 is 2x, and the derivative of 3x is 3)

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

4. Simplify the expression:
f'(x) = (2x^2 + 6x + 4 – (4x^2 + 7x + 3)) / (x^2 + 3x + 2)^2
= (-2x^2 – x + 1) / (x^2 + 3x + 2)^2

Therefore, the derivative of f(x) = (2x + 1) / (x^2 + 3x + 2) is equal to (-2x^2 – x + 1) / (x^2 + 3x + 2)^2.

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