Introduction to The Derivative
Suppose that f is a function whose domain contains the interval (a , b). Let c be a point of (a , b). If the limit
exists then we say that f is differentiable at c and we call the limit the derivative of f at c .
Example 1
Is the function f ( x ) = x ^{2} + x differentiable at x = 2? If it is, calculate the derivative.
Solution 1
We calculate the limit (*), with the role of c played by 2:
We see that the required limit (*) exists, and that it equals 5. Thus the function f ( x ) = x ^{2} + x is differentiable at x = 2, and the value of the derivative is 5.
Math Note: When the derivative of a function f exists at a point c , then we denote the derivative either by f′ (c) or by (d/dx)f(c) = (df/dx)(c) . In some contexts (e.g., physics) the notation f(c) is used. In the last example, we calculated that f′ (2) = 5.
The importance of the derivative is twofold: it can be interpreted as rate of change and it can be interpreted as the slope . Let us now consider both of these ideas.
Suppose that (t) represents the position (in inches or feet or some other standard unit) of a moving body at time t . At time 0 the body is at (0), at time 3 the body is at (3), and so forth. Imagine that we want to determine the instantaneous velocity of the body at time t = c . What could this mean? One reasonable interpretation is that we can calculate the average velocity over a small interval at c , and let the length of that interval shrink to zero to determine the instantaneous velocity. To carry out this program, imagine a short interval [ c , c + h ]. The average velocity of the moving body over that interval is
This is a familiar expression (see (*)). As we let h → 0, we know that this expression tends to the derivative of at c . On the other hand, it is reasonable to declare this limit to be the instantaneous velocity. We have discovered the following important rule:
Let be a differentiable function on an interval ( a , b ). Suppose that (t) represents the position of a moving body. Let c (a, b). Then
′(c) = instantaneous velocity of the moving body at c.
Now let us consider slope. Look at the graph of the function y = f (x) in Fig. 2.6. We wish to determine the “slope” of the graph at the point x = c . This is the same as determining the slope of the tangent line to the graph of f at x = c , where the tangent line is the line that best approximates the graph at that point. See Fig. 2.7.
What could this mean? After all, it takes two points to determine the slope of a line, yet we are only given the point (c, f (c)) on the graph. One reasonable interpretation of the slope at ( c , f (c)) is that it is the limit of the slopes of secant lines determined by (c, f (c)) and nearby points ( c + h , f (c + h)). See Fig. 2.8. Let us calculate this limit:
We know that this last limit (the same as (*)) is the derivative of f at c. We have learned the following:
Let f be a differentiable function on an interval ( a , b ). Let c ( a , b ). Then the slope of the tangent line to the graph of f at c is f′(c).

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