Yet Another Calculus Text, 2007a

26 CHAPTER 1. DERIVATIVES
It now follows that
f(x + dx) − f(x)
=
dx
1
√
x + dx +
√ x
≃
1
√ x +
√ x
= 1
2 √ x .
Thus
f ′ (x) = 1
2 √ x .
For example, the rate of change of y with respect to x when x =9is
f ′ (9) = 1
2 √ 9 = 1 6 .
We will sometimes also write
d
f(x) (1.6.6)
dx
for f ′ (x). With this notation, we could write the result of the previous example
as
d √ 1 x =
dx 2 √ x .
Definition 1.6.2. Given a function f, iff ′ (a) exists we say f is differentiable
at a. Wesayf is differentiable on an open interval (a, b) iff is differentiable at
each point x in (a, b).
Example 1.6.6. The function y = x 2 is differentiable on (−∞, ∞).
Example 1.6.7. The function f(x) = √ x is differentiable on (0, ∞). Note that
f is not differentiable at x =0sincef(0 + dx) =f(dx) is not defined for all
infinitesimals dx.
Example 1.6.8. The function f(x) =x 2 3 is not differentiable at x =0.
1.7 Properties of derivatives
We will now develop some properties of derivatives with the aim of facilitating
their calculation for certain general classes of functions.
To begin, if f(x) = k for all x and some real constant k, then, for any
infinitesimal dx,
f(x + dx) − f(x) =k − k =0. (1.7.1)
Hence, if dx ≠0,
f(x + dx) − f(x)
=0, (1.7.2)
dx
and so f ′ (x) = 0. In other words, the derivative of a constant is 0.
Theorem 1.7.1. For any real constant k,
d
k =0. (1.7.3)
dx
d
Example 1.7.1.
dx 4=0.