Monotonicity rule for the quotient of two functions and its application

Zhen-Hang Yang1, Wei-Mao Qian2, Yu-Ming Chu1

  • 1Department of Mathematics, Huzhou University, Huzhou, 313000 China.

Journal of Inequalities and Applications
|May 30, 2017
PubMed

Related Concept Videos

The Quotient Rule01:30

The Quotient Rule

The quotient rule is a fundamental differentiation technique in calculus used to differentiate functions expressed as a ratio of two differentiable functions. Given a function of the form:Where g(x) and h(x) are both differentiable and h(x) ≠ 0, the derivative of f(x) is given by:Example:The quotient rule is beneficial when differentiating rational functions, trigonometric ratios, and exponential functions. For example, given:applying the quotient rule,This rule is essential in solving...
122
The Product Rule01:24

The Product Rule

In calculus, the Product Rule provides a method for differentiating expressions that are the product of two functions. It states that the derivative of the product of two differentiable functions equals the first function times the rate of change of the second, plus the second function times the rate of change of the first.This rule ensures that the rate of change of the product accounts for the simultaneous variation of both functions.A compelling way to understand the Product Rule is through...
226
Limit Laws II01:26

Limit Laws II

In calculus, limit laws serve as foundational tools for evaluating the behavior of functions as inputs approach specific values. Among these, the laws concerning quotients, powers, and roots are particularly useful in breaking down complex expressions.The Quotient Law allows the limit of a division between two functions to be calculated by dividing their individual limits, provided the limit of the denominator exists and is not zero. For example,The Power Law states that the limit of a function...
301
Indeterminate Forms and L’Hôpital’s Rule01:27

Indeterminate Forms and L’Hôpital’s Rule

Indeterminate forms occur when evaluating limits leads to expressions that cannot be directly interpreted, such as zero divided by zero or infinity divided by infinity. These results do not describe the true behavior of a function near a given point and instead signal that additional analysis is required. L’Hôpital’s Rule provides a reliable method for resolving such ambiguities by replacing the original functions with their derivatives.Core Idea of L’Hôpital’s...
159
Substitution Rule Applied to Definite Integrals01:24

Substitution Rule Applied to Definite Integrals

When evaluating a definite integral whose integrand matches the structure of a composite function, the substitution method provides an efficient way to simplify the calculation. This method is based on reversing the chain rule from differentiation, allowing a complicated expression to be rewritten in a simpler form. When the integrand contains an inner function and its derivative, substitution naturally reduces the complexity of the problem.The core idea of substitution for definite integrals...
114
Slant Asymptotes01:27

Slant Asymptotes

A function's behavior is often guided by asymptotic constraints, where one term dominates another, defining a limiting trend. In the given scenario, the mathematical pattern follows a rational function: a cubic term in the numerator is divided by a squared term in the denominator. This results in a function with distinct characteristics, including an oblique asymptote, critical points, and undefined regions.The function's validity is determined by the denominator, which must be nonzero. This...
185