Calculadora da Regra do Quociente
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
(f/g)' = (f'·g − f·g') / g²
About this calculator
The Quotient Rule Calculator solves derivatives of functions written as a quotient, i.e., one function divided by another. The quotient rule is a fundamental technique in differential calculus, used to find the instantaneous rate of change of ratios of functions. The formula is (f/g)' = (f'·g − f·g') / g², where f and g are differentiable functions. This calculator automates the process, avoiding manual errors and saving time.
To use the tool, enter the functions f(x) and g(x) in the appropriate fields. The calculator computes the derivatives f'(x) and g'(x) and applies the formula step by step. The final result is displayed in simplified form. It is useful for calculus students checking exercises, teachers preparing teaching materials, or professionals working with mathematical models involving ratios of functions.
Common use cases include differentiating rational functions, such as (x²+1)/(x-3), or trigonometric functions, such as sin(x)/cos(x). The quotient rule is essential when the function cannot be simplified to a simpler form before differentiation. Remember that the order of terms in the numerator matters: f'·g minus f·g'.
Cautions: verify that the denominator g(x) is not zero at the point of differentiation, as the derivative does not exist where g(x)=0. Also, the quotient rule can be applied even when the denominator is constant, but in that case the product rule or constant rule is simpler. Always simplify the result if possible.
Frequently asked questions
What if the denominator is zero at the point where I want the derivative?
The derivative does not exist at that point, as the original function is undefined or has a discontinuity. The calculator will indicate an error or undefined result.
Can I use the quotient rule when the numerator or denominator is constant?
Yes, but it is simpler to use the product rule (if the numerator is constant, its derivative is zero) or the constant rule. The calculator accepts any function.
Does the calculator automatically simplify the result?
Yes, the calculator attempts to simplify the final expression by combining terms and factoring when possible.
Do I need to type the function in a specific format?
Use standard notation: use ^ for powers, * for multiplication, and functions like sin(x), cos(x), exp(x), etc. Example: (x^2+1)/(x-3).
Does the quotient rule work for functions with more than one variable?
No, this calculator only handles functions of one variable (x). For partial derivatives, other tools are needed.