Differentiation: Calculating Rate of Change
Differentiation means to find the rate of change of one quantity with respect to another. Differentiation can be applied to many disciplines including physics and economics. In physics, basically there are two aspects of measuring the rate of change. They are displacement of a moving body and the derivative of velocity. In economics, differentiation is applied in various business strategies and helps to find different ways to implant outwitting advantage.
Derivatives & Tangent Lines
A derivative is a measure to find how a function changes with respect to its input. It is of very general interest to know a certain parameter at various instants of time and try to find the rate at which it is changing. Geometrically, the derivative of a function at a point is the slope of the tangent line at the point. A tangent is a straight line that touches a given point on a curve. Slope is a measurement of the degree to which a line is inclined.
- Tangent lines – A pictorial representation and animation of the geometric concept of derivatives.
- Derivatives and tangent lines – Chapter on differentiability and tangent line linear approximation, with tools for demonstration.
- Description about the derivatives – Introduces the calculus concept of derivative and the tangent line.
- Analytical methods – Discussion about slope of nonlinear functions.
- Derivatives applet – The java applet shows pictorial representation of user input function f(x) and its derivative f’(x).
Derivatives & Limits
Sometimes the value of a function never equals a number but continually reaches closer to it. In that case we use the concept of limit. For example, if ‘1/x’ is a function, as the value of ‘x’ increases, the value of the function decreases. It decreases, but never equals zero. Hence, the limit of function ‘1/x’, as x approaches infinity, is zero. The slope of the tangent line, or the derivative, can be determined using a limits, as described further in the following resources.
- Limit Definition of the Derivative – Describes how to find a derivative using the limit definition.
- Derivatives. – A tutorial self-evaluation test on derivatives.
- Limits-examples – Detailed description of derivatives and limits with examples and solved problems of limits.
- Derivatives and limits – The document provides syntax of writing limit problems in Maple.
- Limits and derivatives-introduction – The lab guide teaches how to calculate limits and how to find derivatives.
- Derivatives and limits-quiz – An online PowerPoint presentation for brushing up the concepts of derivatives.
- Limits and derivatives explanation – Limits, continuity, and derivatives are explained with graphical representation.
Differentiation Formulas & Rules
The major rules that define differentiation are the elementary rules like product or Leibniz rule, chain rule, polynomial or elementary power rule, reciprocal rule, and inverse function rule. Some rules for advanced differentiation include the quotient rule and generalized power rule. In every rule, the variable is being taken is x, f(x) denotes the function of x, and c denotes constant. The derivative of a constant function is zero, that is, if c is a constant then, d[c] /dx=0.
- Basic differentiation identities – A tutorial with basic formulas of differentiation.
- Differentiation formulas – An eBook on calculus with chapters on derivatives and application of derivatives.
- Some more formulas – The document contains a list of basic differential formulas.
- Differentiation rules – A collection of concept description, examples, videos, quiz and exercises on derivatives and limits.
Rates of Change Related to Economics
Economics deals with two major functions – cost and revenue. These may deal with average cost, revenue, and marginal profit. When total cost is divided by the total number of units produced, it is known as the average cost. Revenue is defined as the income earned by selling a particular product. Marginal profit is defined as the difference between marginal revenue and marginal cost for generating one additional unit.
- Marginal Analysis – Explains how to make estimates using the derivatives of cost, profit, and revenue functions.
- Applications to Economics – An overview of how calculus is used in economics, including specific functions.
- Economic aspects – The document explains Economic Applications of Implicit Differentiation like substitution of inputs, supply demand equation, implicit differentiation, marginal input of prices etc.
- Economic Application of Derivatives – explanation and description of application of derivatives in economics.
- Application to Economics – The document teaches application of time scale to economics.
- Case Study – The document uses derivatives and other calculus concepts to reach optimal equilibrium among overlapping generations.
Rates of Change Related to Physics
In physics, the rate of change is calculated with respect to time. Two aspects are considered. One is the displacement of a moving body and other is the derivative of velocity. The derivative of the former is velocity of the body whereas the derivative of the latter is acceleration.
- Average Rate of Change – Introduces physical examples of rate of change.
- Instantaneous Velocity – How calculus can be used to determine instantaneous velocity, includes applet.
- Velocity and Acceleration – Description about how to use derivatives to find velocity and acceleration.
- Calculus based physics – The document explains the application of derivatives in science of physics, biology, economics and volumes.
- Derivatives and physics – The comprehensive document explains derivatives in mathematics and physics.
- Simple harmonic motion – Derivatives and limits are used extensively in real-time situations like simple harmonic motion.
Derivative calculators are software calculators that calculate derivatives of any given function. Derivative calculators can be online or an application (.exe file). The user has to input the function and submit it to the software.
- Numerical derivative calculator – Software that can calculate derivative of any function developed by CSERD.
- Derivative calculators download – A collection of links from where the derivative calculators can be downloaded.
- First derivative calculator – Description of first derivative calculators of various developers.
- Online Derivative Calculator – An online derivative calculator with instructions.
- Multivariable Calculus Calculator – Online multivariable derivative calculator.