2.9 The Quotient Rule

The Quotient Rule is a rule in Calculus that states that the derivative of the quotient of two functions is equal to the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. This rule is represented mathematically as: (f(x) / g(x))' = (f'(x) * g(x) - f(x) * g'(x)) / g(x)^2 This rule allows us to find the derivative of a function that is the quotient of two other functions. For example, if we have a function f(x) = x^2 and a function g(x) = 3x, we can use the Quotient Rule to find the derivative of the function h(x) = f(x) / g(x) = x^2 / 3x = x/3. The derivative of h(x) = (f'(x) * g(x) - f(x) * g'(x)) / g(x)^2 = (2x * 3x - x^2 * 1) / (3x)^2 = (6x - x^2) / 9x^2 = 2x / 3x^2 = 2/3x Another example, Find the derivative of f(x)= (x^3-5x+6) / (2x^2+x) Solution: Using the Quotient Rule, the derivative of f(x) = (x^3-5x+6) / (2x^2+x) = (3x^2-5+0) * (2x^2+x) - (x^3-5x+6) * (4x+1) / (2x^2+x)^2 = (3x^2-5) * (2x^2+x) - (x^3-5x+6) * (4x+1) / (2x^2+x)^2 = 6x^4+3x^3-10x^2+5x-4x^4-4x^3-6x^2-6x = 2x^4-x^3-x^2-x It's important to note that the Quotient rule can also be used in combination with other rules, such as the Chain rule, to find the derivatives of more complex functions.

Example 1: Find the derivative of f(x) = x^3 / sin x Solution: Using the Quotient rule, the derivative of f(x) = x^3 / sin x = (3x^2 * sin x - x^3 * cos x) / sin^2 x Example 2: Find the derivative of f(x) = e^x / ln x Solution: Using the Quotient rule, the derivative of f(x) = e^x / ln x = (e^x * ln x - e^x * (1/x)) / ln^2 x Example 3: Find the derivative of f(x) = (x^2+5x+6) / (2x^2+x) Solution: Using the Quotient rule, the derivative of f(x) = (x^2+5x+6) / (2x^2+x) = (2x+5+0) * (2x^2+x) - (x^2+5x+6) * (4x+1) / (2x^2+x)^2

= (2x+5) * (2x^2+x) - (x^2+5x+6) * (4x+1) / (2x^2+x)^2

= 4x^3+2x^2+5x^2+5x-2x^3-10x^2-5x-6

= 2x^3-5x^2-x

Example 4:

Find the derivative of f(x) = x^4 / (x^2+2x+2)

Solution:

Using the Quotient rule, the derivative of f(x) = x^4 / (x^2+2x+2) = (4x^3 * (x^2+2x+2) - x^4 * (2x+2)) / (x^2+2x+2)^2

= (4x^3 * (x^2+2x+2) - x^4 * (2x+2)) / (x^2+2x+2)^2

= (4x^5+8x^4+8x^3-2x^5-4x^4-4x^3) / (x^2+2x+2)^2

= 2x^5+4x^4+4x^3 / (x^2+2x+2)^2

Example 5:

Find the derivative of f(x) = (x^2-2x+1) / (x^2+2x+2)

Solution:

Using the Quotient rule, the derivative of f(x) = (x^2-2x+1) / (x^2+2x+2) = (2x-2+0) * (x^2+2x+2) - (x^2-2x+1) * (2x+2) / (x^2+2x+2)^2

= (2x-2) * (x^2+2x+2) - (x^2-2x+1) * (2x+2) / (x^2+2x+2)^2

= 2x^3+4x^2+4x-2x^3-4x^2-4x-2x^2-4x-2

= -x^2-2x

Example 6:

Find the derivative of f(x) = (x^2+2x+1) / (x^3+3x^2+3x+1)

Solution:

Using the Quotient rule, the derivative of f(x) = (x^2+2x+1) / (x^3+3x^2+3x+1) = (2x+2+0) * (x^3+3x^2+3x+1) - (x^2+2x+1) * (3x^2+6x+3) / (x^3+3x^2+3x+1)^2

In conclusion, the Quotient Rule is a fundamental rule in Calculus that allows us to find the derivative of a function that is the quotient of two other functions. The rule is represented mathematically as (f(x) / g(x))' = (f'(x) * g(x) - f(x) * g'(x)) / g(x)^2 and it is important to note that it can be used in combination with other rules such as the Chain Rule to find the derivatives of more complex functions. The examples provided in this guide demonstrate how the Quotient Rule can be applied to different functions and the steps required to solve for the derivative. Understanding and being able to apply the Quotient Rule is crucial for advanced Calculus and mathematical research.