Page 45 - 2024-calc4e-SE proofs-4e.indd

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

x
Section 2.3 • The Derivative of a Polynomial Function; The Derivative of y = e and y = ln x 195

d
2
3
(a) f (x) = 5 · x , so f (x) = 5 x 3 = 5 · 3x = 15x 2
′
dx
1 1 d 1
2
1
2
′
(b) g(u) = − · u , so g (u) = − · u = − · 2u = −u
2 2 du 2
d
4 3 4 3 4 2 4 2
′
(c) u(x) = π x , so u (x) = π · x = π · 3x = 3π x
dx
© 2024 BFW Publishers PAGES NOT FINAL - For Review Purposes Only - Do Not Copy.
↑
π is a constant
R
NOW WORK Problem 31 and AP Practice Problem 10.
3 Differentiate the Sum and the Difference of Two Functions
We can find the derivative of a function that is the sum of two functions whose
derivatives are known by adding the derivatives of each function.

THEOREM Sum Rule
If two functions f and g are differentiable and if F(x) = f (x) + g(x), then F is
IN WORDS The derivative of the sum of two differentiable and
differentiable functions equals the sum of
′
′
′
their derivatives. That is, ( f + g) = f + g . ′ F (x) = f (x) + g (x)
′
′
Proof If F(x) = f (x) + g(x), then
F(x + h) − F(x) = [ f (x + h) + g(x + h)] − [ f (x) + g(x)]
= [ f (x + h) − f (x)] + [g(x + h) − g(x)]

So, the derivative of F is
[ f (x + h) − f (x)] + [g(x + h) − g(x)]
F (x) = lim
′
h→0 h
f (x + h) − f (x) g(x + h) − g(x)
= lim + lim The limit of a sum is
h→0 h h→0 h
the sum of the limits.
= f (x) + g (x)
′
′
In Leibniz notation, the Sum Rule takes the form
d d d
[ f (x) + g(x)] = f (x) + g(x)
dx dx dx

EXAMPLE 4 Differentiating the Sum of Two Functions

2
Find the derivative of f (x) = 3x + 8.
Solution
2
Here f is the sum of 3x and 8. So, we begin by using the Sum Rule.
d d d d
′ 2 2 2
f (x) = (3x + 8) = (3x ) + 8 = 3 x + 0 = 3 · 2x = 6x
dx ↑ dx dx ↑ dx ↑
Sum Rule Constant Multiple Simple
Rule Power Rule
R
NOW WORK Problem 7 and AP Practice Problem 6.
© 2024 BFW Publishers PAGES NOT FINAL
For Review Purposes Only, all other uses prohibited
Do Not Copy or Post in Any Form.