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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

192 Chapter 2 • The Derivative and Its Properties

dy
Leibniz notation may be written in several equivalent ways as
dx
dy d d
= y = f (x)
dx dx dx
d
where is an instruction to find the derivative of the function y = f (x) with respect
dx
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to the independent variable x.
In operator notation Df (x), D is said to operate on the function, and the result
is the derivative of f . To emphasize that the operation is performed with respect to the
independent variable x, it is sometimes written Df (x) = D x f (x).
We use prime notation or Leibniz notation, or sometimes a mixture of the two,
depending on which is more convenient. We do not use the notation Df (x) in this book.

1 Differentiate a Constant Function
y

A See Figure 23. Since the graph of a constant function f (x) = A is a horizontal line,
Slope  0 the tangent line to f at any point is also a horizontal line, whose slope is 0. Since the
derivative is the slope of the tangent line, the derivative of f is 0.
x
Figure 23 f (x) = A THEOREM Derivative of a Constant Function
If f is the constant function f (x) = A, then

f (x) = 0
′
That is, if A is a constant, then

d
A = 0
dx
IN WORDS The derivative of a constant is 0.

Proof If f (x) = A, then its derivative function is given by

f (x + h) − f (x) A − A
′
f (x) = lim = lim = 0
h→0 h h→0 h
↑ ↑
The definition of a f (x) = A
derivative, Form (2) f (x + h) = A

EXAMPLE 1 Differentiating a Constant Function
√ 1
′
′
(a) If f (x) = 3, then f (x) = 0 (b) If f (x) = − , then f (x) = 0
2
d d
(c) If f (x) = π, then π = 0 (d) If f (x) = 0, then 0 = 0
dx dx
2 Differentiate a Power Function; the Simple Power Rule
y
2 n
Next we analyze the derivative of a power function f (x) = x , where n ≥ 1 is an integer.
1
When n = 1, then f (x) = x is the identity function and its graph is the line y = x,
as shown in Figure 24.
2 1 1 2 x
′
1 The slope of the line y = x is 1, so we would expect f (x) = 1.
2
d f (x + h) − f (x) (x + h) − x h
′ x = lim = lim = lim = 1
Proof f (x) =
Figure 24 f (x) = x dx h→0 h h→0 h h→0 h
↑
f (x) = x, f (x + h) = x + h
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