quotient rule
The quotient rule is a formula used to differentiate functions that are in the form of a quotient, or a division, of two functions
The quotient rule is a formula used to differentiate functions that are in the form of a quotient, or a division, of two functions. It is especially useful when differentiating complicated equations involving fractions.
The quotient rule states that if we have a function in the form of f(x) = g(x) / h(x), where g(x) and h(x) are both functions of x, 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
Let’s break down the quotient rule step by step and understand how it is derived:
1. Start with the original function f(x) = g(x) / h(x).
2. To find the derivative of f(x), we can rewrite it as a difference of two fractions using a common denominator:
f(x) = g(x) * (1 / h(x))
3. Now, we need to differentiate both g(x) and 1/h(x).
The derivative of g(x), denoted as g'(x), is obtained using standard differentiation rules.
The derivative of 1/h(x), which is (d/dx)(1/h(x)), can be found by applying the chain rule:
(d/dx)(1/h(x)) = -(1 / [h(x)]^2) * h'(x)
4. Substitute the derivatives back into the original formula of f'(x):
f'(x) = g'(x) * (1 / h(x)) – g(x) * (1 / [h(x)]^2) * h'(x)
5. Simplify the formula:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / [h(x)]^2
So, the final formula for the derivative of a quotient function, or the quotient rule, is:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / [h(x)]^2
By using the quotient rule, you can differentiate functions that contain divisions and fractions and determine how their rates of change vary with respect to x.
More Answers:
Simplifying Algebraic Expressions: The Sum and Difference Rules ExplainedExploring Higher Order Derivatives: Analyzing Function Behavior and Curvature
A Step-by-Step Guide to Applying the Product Rule in Calculus for Finding Derivatives of Function Products