Pg 225

h(t) 4.9t 2 1.5t 17
h′(t) lim
h → 0
lim
h → 0
lim
h → 0
lim
h → 0
lim
h → 0
lim
h → 0
lim
f (t h) f (t)
h
[4.9(t h) 2 1.5(t h) 17] [4.9t 2 1.5t 17]
h
4.9(t 2 2th h 2 ) 1.5(t h) 17 [4.9t 2 1.5t 17]
h
4.9t 2 9.8th 4.9h 2 1.5t 1.5h 17 4.9t 2 1.5t 17
h
9.8th 4.9h 2 1.5h
h
h 1 (9.8t 4.9h 1.5)
h
1
20.0
(9.8t 4.9h 1.5) 10.0
h → 0
c′(t) = 1.5t
0.0
d′(t) = 0
9.8t 4.9(0) 1.5
9.8t 1.5
The derivative of the original
function, f (t), is the sum of the
derivatives of the component
functions, b(t), c(t), and d(t).
f (t) b(t) c(t) d(t)
4.9t 2 1.5t 17
f ′(t) b′(t) c′(t) d′(t)
9.8t 1.5
Value of Derivative
–10.0
–20.0
–30.0
0.4 0.8 1.2 1.6 2.0 2.4
Time (s)
f ′(t) = b′(t) + c′(t) + d′(t)
f′(t) = b′(t) + c′(t) + d′(t)
= –9.8t + 1.5
b′(t) = – 9.8t
The derivative of a sum is
the sum of the derivatives.
The Sum Rule
If h(x) f (x) g(x) and f and g are both differentiable, then
h′(x) f ′(x) g′(x). In Leibniz notation,
d
d
[ f (x) g(x)] d
[ f (x)] d
[g(x)].
d
d
x
x
x
Proof
Let F(x) f(x) g(x).
h ′(x) lim
h → 0
lim
h → 0
lim
h → 0
F(x h) F(x)
h
f (x h) f (x)
h
lim
h → 0
f ′(x) g′(x)
[ f (x h) g(x h)] [ f (x) g(x)]
h
[f (x h) f (x)] [g(x h) g(x)]
h
lim
h → 0
g(x h) g(x)
h
Rearrange the numerator.
Rewrite using the
sum rule for limits.
226 CHAPTER 3 RATES OF CHANGE IN POLYNOMIAL FUNCTION MODELS