The quotient rule is a differentiation formula that simplifies the process of finding the derivative of a quotient of two functions. It involves four key concepts: the dividend function, the divisor function, the differential of the dividend, and the differential of the divisor. Understanding these concepts and applying the rule correctly enables one to accurately determine the derivative of a quotient function.
Quotient Rule Simplified: A Step-by-Step Guide
The quotient rule in calculus is a technique used to calculate the derivative of a fraction. It’s a bit more involved than other differentiation rules, but with a clear understanding of its structure, you can master it in no time.
Formula:
$$\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{g(x)f'(x) – f(x)g'(x)}{[g(x)]^2}$$
Steps:
-
Label the functions: Let u = f(x) and v = g(x).
-
Apply the rule:
$$ \frac{du}{dx} \cdot v – u \cdot \frac{dv}{dx} $$
$$[g(x)f'(x) – f(x)g'(x)]$$ -
Square the denominator: $$[g(x)]^2$$
Example:
Find the derivative of y = (x^2 – 1)/(x + 2)
- u = x^2 – 1
- v = x + 2
- du/dx = 2x
- dv/dx = 1
Using the quotient rule:
$$\frac{dy}{dx} = \frac{(x+2)(2x) – (x^2-1)(1)}{(x+2)^2}$$
$$= \frac{4x^2 + 4x – x^2 + 1}{(x+2)^2}$$
$$= \frac{3x^2 + 4x + 1}{(x+2)^2}$$
Table of Common Quotients:
| Quotient | Derivative |
|---|---|
| (x^2 + 1)/(x – 1) | (2x-1)/(x-1)^2 |
| (x – 1)/(x^2 + 1) | (2x)/(x^2 + 1)^2 |
| (sin x)/x | (x cos x – sin x)/x^2 |
| (e^x)/ln x | e^x/(ln x)^2 |
Question 1:
What is the “quotient rule low d high” for derivatives?
Answer:
The “quotient rule low d high” states that the derivative of a quotient of two functions, f(x) = g(x)/h(x), is given by:
f'(x) = [(hd’-gd’)h] / h^2
Where d’ denotes the derivative with respect to x.
Question 2:
How is the quotient rule derived?
Answer:
The quotient rule can be derived using the limit definition of the derivative:
lim (h->0) [g(x+h)/h(x+h) – g(x)/h(x)] / h
= lim (h->0) [(h(x)g'(x) – g(x)h'(x)) / h(x)h(x+h)]
= [(hd’-gd’)h] / h^2
Question 3:
What are the conditions for applying the quotient rule?
Answer:
The quotient rule can be applied under the following conditions:
- Both g(x) and h(x) must be differentiable at x.
- h(x) must not be equal to 0 at x.
- The limit of the quotient as h approaches 0 must exist.
Well, there you have it, folks! I hope you enjoyed this quick dive into the quotient rule. It’s a powerful tool that can help you tackle even the trickiest of derivatives. If you have any further questions or just want to nerd out about calculus, feel free to drop me a line. I’m always happy to chat about the wonders of math. Thanks for reading, and be sure to visit again soon for more math adventures!