1.3. THE DERIVATIVE OF A FUNCTION AT A POINT
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1
2
-4
-2
y = x − x 2
m = f ′ (2)
Figure 1.13: The tangent line to y = x − x 2 at the point (2, −2).
algebra. Expanding and distributing in the numerator,
f
′ (2) = lim
h→0
2 + h − 4 − 4h − h 2 + 2
h
.
Combining like terms, we have
f
′ (2) = lim
h→0
−3h − h 2
h
.
Next, we observe that there is a common factor of h in both the numerator and denominator, which allows us to simplify and find that
f
′ (2) = lim
h→0
(−3 − h).
Finally, we are able to take the limit as h → 0, and thus conclude that f ′ (2) = −3.
Now, we know that f ′ (2) represents the slope of the tangent line to the curve y = x − x 2
at the point (2, −2); f ′ (2) is also the instantaneous rate of change of f at the point (2, −2).
Graphing both the function and the line through (2, −2) with slope m = f ′ (2) = −3, we
indeed see that by calculating the derivative, we have found the slope of the tangent line
at this point, as shown in Figure 1.3.
The following activities will help you explore a variety of key ideas related to derivatives.