Derivative of Rational Function
The derivative of a rational function is found by applying the quotient rule, which involves differentiating the numerator and denominator separately and then combining the results. This process allows engineers and technologists to analyze the rate of change of a rational function, which is crucial for understanding how variables interact in various technological and engineering applications.
3 Key excerpts on "Derivative of Rational Function"
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...8 Limits, Derivatives, Integrals, and Differential Equations Most functions of practical interest – in that they can be used to simulate physicochemical phenomena, are intrinsically continuous; this means that they evolve smoothly along their independent variable, i.e. the value taken by the function at a given value of its independent variable coincides with what would be expected from the trends in the vicinity of said value for the latter variable. This calls for the concept of limit – based on realization that an infinite sequence of values taken by a function may be such that every element will never go beyond a finite threshold; said topic is particularly relevant when seeking asymptotic behaviors of functions – since they tend to take forms simpler than the original functions themselves; or when quantifying the tendency of evolution in the neighborhood of some point via the concept of derivative (or ratio of small variations of the function to its independent variable). Derivatives find their widest applicability when searching for optima – one of the most seminal goals of process engineers. In fact, local maxima of a given function are described by nil values of the corresponding derivative if not subjected to otherwise (externally) imposed physical constraint(s); decision on the type of optima would then come at the expense of higher order derivatives – thus extending the concept of differentiation of a function itself to differentiation of a previous derivative. Once a function (seen as an algebraic operator) is proposed, one may in principle define its inverse; this supports the common strategy to solve an algebraic equation. A similar rationale may be applied to the derivative – in which case the concept of integral arises...
eBook - ePub- Adil H. Mouhammed(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...CHAPTER TWO Derivatives and Applications Economic decisions are based on marginal analysis. For example, the monopolist's best level of output is determined by equating marginal cost and marginal revenue. To find the marginal cost and revenue, total cost and revenue functions must be differentiated with respect to the output level. Similarly, derivatives can be used in many applications in business and economics. For this reason, the rules of differentiation are outlined in this chapter, and many applications are provided. The Concept of Derivative The derivative of a function measures the rate of change of the dependent variable y with respect to the independent variable x--the slope of the function. That is, the derivative indicates the impact of a small change in x on y. For example, suppose the dependent variable y is the quantity supplied by a producer, and x is the price of that product. Mathematically, the function is written as y = f(x). Now, if the price x changes by a very small amount (dx), the quantity supplied will change by a very small amount (dy) as well. These small changes, dx and dy, are called the differential of x and y, respectively. After these changes, the new magnitude of the two variables becomes (y + dy) and (x + dx). And dy/dx is called the derivative of y with respect to x. In other words, dy/dx shows the changes in y per unit change in x. The process of finding the derivative is called the differentiation process. If a given function is a univariate function, such as the above, the following rules of differentiation (Glaister 1984; Chiang 1984; Ostrosky and Koch 1986) are applied: Rule 1: Derivative of a Constant function If y = f(x) = k, where k is a constant, then dy/dx = 0 Example 1: Differentiate y = f(x) = 30. Solution: dy/dx = 0 Example 2: If the fixed cost of a product q is FC = 20, then the derivative of the fixed cost with respect to q is d(FC)/dq = 0...
eBook - ePub- Adil H. Mouhammed(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Chapter Two Derivatives and Applications Economic decisions are based on marginal analysis. For example, the monopolist’s best level of output is determined by equating marginal cost and marginal revenue. To find the marginal cost and revenue, total cost and revenue functions must be differentiated with respect to the output level. Similarly, derivatives can be used in many applications in business and economics. For this reason, the rules of differentiation are outlined in this chapter, and many applications are provided. The Concept of Derivative The derivative of a function measures the rate of change of the dependent variable y with respect to the independent variable x--the slope of the function. That is, the derivative indicates the impact of a small change in x on y. For example, suppose the dependent variable y is the quantity supplied by a producer, and x is the price of that product. Mathematically, the function is written as y = f (x). Now, if the price x changes by a very small amount (dx), the quantity supplied will change by a very small amount (dy) as well. These small changes, dx and dy, are called the differential of x and y, respectively. After these changes, the new magnitude of the two variables becomes (y + dy) and (x + dx). And dy/dx is called the derivative of y with respect to x. In other words, dy/dx shows the changes in y per unit change in x. The process of finding the derivative is called the differentiation process. If a given function is a univariate function, such as the above, the following rules of differentiation (Glaister 1984; Chiang 1984; Ostrosky and Koch 1986) are applied: Rule 1: Derivative of a Constant Function If y = f(x) = k, where k is a constant, then dy/dx = 0 Example 1: Differentiate y = f(x)= 30. Solution: dy/dx = 0 Example 2: If the fixed cost of a product q is FC = 20, then the derivative of the fixed cost with respect to q is d(FC)/dq = 0...
