Derivatives are all about instantaneous rate of change. Therefore, when we interpret the rate of a function given the value of its derivative, we should always For linear functions, we have seen that the slope of the line measures the average rate of change of the function and can be found from any two points on the line. Relative Rate of Change. The relative rate of change of a function f(x) is the ratio if its derivative to itself, namely. R(f(x))=(f^'(x)). SEE ALSO: Derivative, Function, If this limit exists, we call it the derivative of f at x=a. In the following video, we use this definition to compute the instantaneous velocity at time t=2 DERIVATIVES AND RATES OF CHANGE. EXAMPLE A The flash unit on a camera operates by storing charge on a capaci- tor and releasing it suddenly when if a changing quantity is defined by a function, we can differentiate and evaluate the derivative at given values to determine an instantaneous rate of change: Understand that the derivative is a measure of the instantaneous rate of change of a function. Differentiation can be defined in terms of rates of change, but what

## Same as the value of the derivative at a particular point. For a function, the instantaneous rate of change at a point is the same as the slope of the tangent line. That

The instantaneous rate of change of a function is an idea that sits at the foundation of calculus. It is a generalization of the notion of instantaneous velocity and Second derivatives (and third derivatives, and so on) are also functions ! Each one tells us about the rate of change of the previous function in this pyramid The derivative of a composite function turns out to be the product of the derivatives of the separate functions. The exact statement is given in the following box. To Definition. Let s(t) be the position function, then the instantaneous velocity at v(t) is the derivative of the position function. v(t) = s'(t) While estimates of the instantaneous rate of change can be found using values and times, an exact calculation requires using the derivative function. This rate of The rate of change of a function varies along a curve, and it is found by taking the first derivative of the function. The derivative, , of a function y(x) is the rate of Measuring Rates of Change. This module begins a very gentle introduction to the calculus concept of the derivative. The first lesson, "This is About the Derivative

### 2 - Find a formula for the rate of change dA/dt of the area A of a square whose side x centimeters changes at a rate equal to 2 cm/sec. 3 - Two cars start moving from the same point in two directions that makes 90 degrees at the constant speeds of s1 and s2.

We want to find the average rate of change of (handfuls of trail mix) with respect to feet. The independent variable goes from 0 ft to 200 ft. The dependent variable goes from 0 handfuls to 3 handfuls. The average rate of change is The population growth rate is the rate of change of a population and consequently can be represented by the derivative of the size of the population. Definition If P ( t ) P ( t ) is the number of entities present in a population, then the population growth rate of P ( t ) P ( t ) is defined to be P ′ ( t ) . The derivative, f0(a) is the instantaneous rate of change of y= f(x) with respect to xwhen x= a. When the instantaneous rate of change is large at x 1, the y-vlaues on the curve are changing rapidly and the tangent has a large slope. When the instantaneous rate of change ssmall at x 1, the y-vlaues on the Section 4-1 : Rates of Change The purpose of this section is to remind us of one of the more important applications of derivatives. That is the fact that \(f'\left( x \right)\) represents the rate of change of \(f\left( x \right)\). 4. The Derivative as an Instantaneous Rate of Change. The derivative tells us the rate of change of one quantity compared to another at a particular instant or point (so we call it "instantaneous rate of change"). This concept has many applications in electricity, dynamics, economics, fluid flow, population modelling, queuing theory and so on.