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 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.

  • Derivatives. – A tutorial self-evaluation test on derivatives.
  • Limits-examples – Detailed description of derivatives and limits with examples and solved problems of limits.

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.

  • 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.
  • Economic aspects – The document explains Economic Applications of Implicit Differentiation like substitution of inputs, supply demand equation, implicit differentiation, marginal input of prices etc.
  • 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.

  • Calculus based physics – The document explains the application of derivatives in science of physics, biology, economics and volumes.
  • Simple harmonic motion – Derivatives and limits are used extensively in real-time situations like simple harmonic motion.

Derivative Calculators

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.