                   # Derivatives

## The Derivative at a Point

The derivative of a function at a point is its instantaneous rate of change at that point. It is also the slope of the tangent line to the graph of the function at that point (provided that the tangent line exists and is not vertical). Given a function, we can use the notation , to denote the derivative of f  with respect to t at the point t = a or the instantaneous rate of change of f at t = a.

Example 1

Consider the function C(t )= -.0217t ²- 0.9205t + 13.0983, which can be used to model the population (in thousands) of Chattahoochee County, Ga in the year 2000 + t. Interpret the following statements in the context of this situation.

(a) C′(-25)= 0.165 thousand people per year

(b) C′(0)= -0.921 thousand people per year

(c) The derivative of C with respect to t is -0.487 thousand people per year when

t = -10.

### Solution

(a) This statement says that the population of Chattahoochee County was increasing by 165 people per year at the beginning of 1975.

(b) This means that in the beginning of the year 2000, the population of Chattahoochee County was decreasing by 921 people per year.

(c) The third statement tells us that the population of Chattahoochee County was decreasing by 487 people per year in the beginning of 1990.

Notice that all of these statements tell us something about how the population of Chattahoochee County was changing at a particular time. They do not, however, tell us what the population was at that time. We can get that information from the function C(t).

### Example 2

Figure 1 below shows the graph of a function f(t) and its tangent line at the point

t = -1. Estimate the values of f(-1) and f′(-1). Figure 1

### Solution

Since f(-1) is the y-coordinate of the point on the graph of f  where t = -1, we see that f(-1) = 2. To find f′(-1), we must estimate the slope of the tangent line at the point t = -1. We choose two points on the tangent line, (-2.5,0) and (-1,2), and calculate its slope: So, f′(-1) ≈ 1.33.

### Example 3

If f(x) = x²-2x+1, estimate the value of f′(2) .

### Solution

Since f′(2) is the slope of the tangent line to the graph of f(x) at the point x = 2, we can use the graphical method to find f′(2).That is we graph the function and zoom-in on the point x = 2 until the graph looks like its tangent line. Figure 2 shows a close-up view of the part of the parabola containing the point x = 2. Figure 2

We see that a good estimate of the slope at the point x = 2 can be obtained by choosing the two points (1.9, 0.8) and (2,1). Our estimate is This means that the slope of the tangent line to the graph of f(x) = x²-2x+1 at the point (2,1) is approximately 2.0.

Note: You could choose any two points on the above graph and get an answer that is close to 2.0.

## The Derivative Function

Since the derivative of a function at a point is the slope or rate of change of the function at that point, the derivative takes on exactly one value for each different value of x in the domain of f (as long as the tangent line exists and is non-vertical at that point). Therefore, the derivative is also a function. Its domain is the set of all numbers in the domain of f for which the graph of f has a non-vertical tangent line. There are several different notations for the derivative of a function. Three of the most commonly used notations are given below.

If y  = f(x) is a function, then

(1) f′(x), read  f prime of x, denotes the derivative of f with respect to x;

(2) denotes the derivative of y with respect to x;

(3) denotes the derivative of f with respect to x.

We can use the following derivativedefinition to find f′(x) for any given function f(x):

### Derivative Definition ### Example 1

Find the derivative of the function ### Solution Substituting into the derivative formula gives,

### Basic Derivative Rules

The following differentiation formulas can be used to find the derivative of many basic algebraic functions.

1. ### Derivative of a Linear Function

If ### Example

Find the derivative of the linear function ### Solution 1. ### The Constant Rule ### Example

Find the derivative of the constant function ### Solution 1. ### The Power Rule ### Example

Find the derivative of each of the functions ### Solution

(i) ; (ii) ; (iii) 1. ### The Constant Multiplier Rule

If ### Example

Find the derivative of the function ### Solution

Using the Constant Multiplier Rule and the Power Rule, we get 1. ### The Sum Rule Example

Find the derivative of the function ### Solution

Applying the Sum, Constant Multiplier, Power, and Constant Function Rules gives 1. ### The Product Rule

If ### Example

Find the derivative of the function ### Solution 1. ### The Quotient Rule

If ### Example

Find the derivative of the function ### Solution 1. ### The Chain Rule ### Example

Find the derivative of the function ### Solution

Using the chain rule gives   