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Applied calculus topic summary: introduction to the derivative 
Average Rate of Change of f(x) over [a, b]: Difference Quotient
The average rate of change of f(x) over the interval [a, b] is
Alternative Formulation: Average Rate of Change of f(x) over [a, a+h] (Replace b above by a+h.) The average rate of change of f(x) over the interval [a, a+h] is
Units: The units of the average rate of change are units of f per unit of x. 
Example
Let f(x) = 2x^{2}  4x + 1. Then the average rate of change of f(x) over the interval [2, 4] is
Interpretation: If, say f(x) represents the annual profit of your company (in millions of dollars) and x represents the year since January 2003, then the units of measurement of the average rate of change are millions of dollars per year. Thus, your company made an average annual profit of $8 million per year over the period January 2005 (t = 2) to January 2007 (t = 4). Use the handy little utility below to compute the average change of the above function f(x) over other intervals. Enter the xcoordinates (a and b in the formula), leave everything else blank, and press "Compute." (You can also change the function to anything you like, using standard technology formatting.)
You can also use the function evaluator to compute average rates of change. 

Instantaneous Rate of Change of f(x) at x = a: Derivative
The instantaneous rate of change of f(x) at x = a is given by taking the limit of the average rates of change (computed by the difference quotient) as h approaches 0.
Note:
Since f'(x) is a limit, it may or may not exist. That is, the quantities [f(x+h)  f(x)]/h may or may not approach a fixed number as h approaches zero. If everything works out fine and the limit exists, then we say that f is differentiable at x. Otherwise, we say that f is not differentiable at x. On this page, we summarize three ways of obtaining the derivative of a function at a point: numerical, graphical, and algebraic. 
Examples
Let f(x) = 2x^{2}  4x + 1, as above. Then the instantaneous change of f(x) at x = 2 is
Interpretation


Numerical Approach
To compute an approximate value of f'(a) (for a given value of a) numerically, one can use either:
Using a Table
A Quick Approximation
Another Quick Approximation: Balanced Difference Quotient
Derivative Calculator (Balanced Difference Quotient Approximation)

Examples
Continuing with the example f(x) = 2x^{2}  4x + 1, let us compute an approximate value of f'(2). Using a Table: The difference quotient (with a = 2) is
As h gets smaller, we see that the value gets closer and closer to 4, so we conclude
Using A Quick Approximation (Forward Difference Quotient): We use the formula (with a = 2)
Notice that the "quick approximation" method does not give the exact answer, but the balanced difference quotient will in this case (see opposite). 

Geometric Approach: The Derivative as Slope
Secant and Tangent Lines
This is also the formula for the average rate of change of f over [a,a+h]. So, The slope of the tangent line through the point on the graph of f where x = a is obtained by moving the point Q closer to P; in other words, by letting h approach 0:
This is also the formula for the instantaneous rate of change of f at the point a. So, We can approximate the slope of the tangent through the point where x = a by using the balanced difference quotient,
Zooming In
Extra Topic: Graph of the Derivative

Examples
Continuing with the example f(x) = 2x^{2}  4x + 1,
Here is the graph with these two lines shown: Zooming In Here is an illustration of zooming in to a point on a graph where x = 0.75. 

Computing the Derivative Algebraically
To compute the derivative of a function algebraically, proceed as follows.

Example
Going back to our first example, f(x) = 2x^{2}  4x + 1, let us now calculate f'(x) algebraically by following the steps in the adjacent window.
Thus, f'(x) = 4x4. Go to the tutorial on average rates of change for practice in computing average rates of change algebraically (what we did above up to the last step), or to tutorial on computing the derivative algebraically and scroll down to the box called "Computing the Derivative Algebraically." The "Help" button brings up the complete solution. 

Velocity
For an object moving in a straight line with position s(t) at time t, the average velocity from time t to time t+h is given by the difference quotient
The instantaneous velocity at time t is given by

Examples
Suppose the position of a moving object is given by s(t) = t^{2 }2t+4 milesat time t hours. Then its velocity at time t is given by s'(t) = 2t2 miles per hour.Thus, for example, its velocity at time t = 3 hours is s'(3) = 4 miles per hour. 