Quotient Rule
The quotient rule is a formula used to find the derivative of a function that is the quotient of two other functions
The quotient rule is a formula used to find the derivative of a function that is the quotient of two other functions. It is particularly useful when differentiating functions that involve a division operation.
The quotient rule states that if you have a function f(x) = g(x) / h(x), where g(x) and h(x) are both differentiable functions, 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
In words, the quotient rule tells us that to find the derivative of a quotient function, we take the derivative of the numerator and multiply it by the denominator, then subtract the product of the numerator and the derivative of the denominator, all divided by the square of the denominator.
To apply the quotient rule, follow these steps:
1. Identify the numerator (g(x)) and denominator (h(x)) of the function you want to differentiate.
2. Differentiate the numerator to find g'(x), and differentiate the denominator to find h'(x).
3. Plug these values into the quotient rule formula: f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
4. Simplify the expression if possible.
Here’s an example to demonstrate how the quotient rule is used:
Given the function f(x) = (2x^2 + 3x) / (x – 1), we can find its derivative using the quotient rule.
Step 1: Identify the numerator and denominator.
g(x) = 2x^2 + 3x
h(x) = x – 1
Step 2: Differentiate the numerator and denominator.
g'(x) = 4x + 3
h'(x) = 1
Step 3: Apply the quotient rule formula.
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
= ((4x + 3) * (x – 1) – (2x^2 + 3x) * 1) / (x – 1)^2
= (4x^2 – 4x + 3x – 3 – 2x^2 – 3x) / (x – 1)^2
= (2x^2 – 4x – 3) / (x – 1)^2
So, the derivative of f(x) = (2x^2 + 3x) / (x – 1) is f'(x) = (2x^2 – 4x – 3) / (x – 1)^2.
Remember to always simplify and check for any further algebraic simplification you can do to make the expression as clean and simple as possible.
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