Calculus- Early Transcendentals, 2021a

130 Derivatives
Exercise 4.2.5 Find a value for a so that the graph of f (x)=x 2 + ax − 3 has a horizontal tangent line at
x = 4.
4.3 Derivative Rules
Using the definition of the derivative of a function is quite tedious. In this section we introduce a number
of different shortcuts that can be used to compute the derivative. Recall that the definition of derivative is:
Given any number x for which the limit
f ′ (x)=lim
h→0
f (x + h) − f (x)
h
exists, we assign to x the number f ′ (x).
Next, we give some basic derivative rules for finding derivatives without having to use the limit definition
directly.
Theorem 4.14: Derivative of a Constant Function
Let c be a constant, then d dx (c)=0.
Proof. Let f (x)=c be a constant function. By the definition of derivative:
f ′ f (x + h) − f (x) c − c
(x)=lim
= lim = lim 0 = 0.
h→0 h
h→0 h h→0
♣
Example 4.15: Derivative of a Constant Function
The derivative of f (x)=17 is f ′ (x)=0 since the derivative of a constant is 0.
Theorem 4.16: The Power Rule
If n is a positive integer, then d dx (xn )=nx n−1 .
Proof. We use the formula:
x n − a n =(x − a)(x n−1 + x n−2 a + ···+ xa n−2 + a n−1 )
which can be verified by multiplying out the right side. Let f (x) =x n be a power function for some
positive integer n. Then at any number a we have:
f ′ f (x) − f (a) x n − a n
(a)=lim
= lim
x→a x − a x→a x − a = lim
x→a (xn−1 + x n−2 a + ···+ xa n−2 + a n−1 )=na n−1 .