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The Derivative
The Tangent Line
Let two points on the graph of a function, f(x), be (a, f(a)) and (a + h, f(a + h)). The line passing
through these points is called the secant line (see figure 1) and its slope is equal to the average
rate of change between the two points.
Slope of secant line = average rate of change =
f ( a h ) f ( a )
h
If we let h approach zero, the point (a + h, f(a + h)) will get closer and closer to the point
(a, f(a)), as shown in figure 1. This would then result in giving us the instantaneous rate of
change where x = a, which is what is called the slope of the tangent line. Therefore, a tangent
line is a line that touches the graph of a function at only one point provided that the limit as h
approaches zero of the difference quotient exists.
Figure 1
This slope and the point (a, f(a)) can then be substituted into the point-slope form of a line to
determine the equation of the tangent line.
Point-slope form of a line
y โ y 1 = m(x โ x 1 )
where m = slope of tangent line and (x 1 , y1 ) = the point (a, f(a))
Example 1: Find the slope of the tangent line to the graph of f(x) = x
2
Find the equation of the tangent line.
Solution:
To determine the slope of the tangent you would begin by finding f(1) and
f(1 + h).
f(x) = x
2
f(1) = (1)
2
f(1) = 1 + 3 โ 4
f(1) = 0
f(x) = x
2
f(1 + h) = (1 + h)
2
f(1 + h) = 1 + 2h + h
2
f(1 + h) = h
2
Now substitute f(1) and f(1 + h) into the formula for the slope of the tangent line.
Slope of tangent line = 0
lim h
f h f
โ h
2
0 0
2
0
0
0
lim lim
lim
lim
lim( 5)
h h
h
h
h
f h f h h
h h
h h
h
h h
h
h
โ โ
โ
โ
โ
Next, substitute the calculated slope and point into the point-slope form of a line.
(a, f(a)) = (1, 0) and m = 5
y โ y 1 = m(x โ x 1 )
y โ 0 = 5(x โ 1)
y = 5x โ 5
The equation of the tangent line to f(x) at the point (1, 0) is y = 5x โ 5.
Example 2 (Continued):
Step 3: Divide by โhโ
2 ( ) ( ) 4 2 5
f x h f x xh h h
h h
h x h
h
x h
Step 4: Find the derivative
0
0
lim
lim 4 2 5
h
h
f x h f x f x h
x h
x
x
โ
โ
There are several situations where the derivative of a function will not exist. These included
- At sharp points or corners
- At points where the function is discontinuous
- At vertical asymptotes
- At points where the slope of the tangent line is undefined (vertical tangent line)
- At points where the function is undefined
