Math 100 Worksheet 2
September 26, 2020
Question 1. Use the definition of derivative to find f 0 (0) where
2
x sin x1
if x 6= 0,
f (x) =
0
if x = 0.
Question 2. Use the limit definition of derivative to prove the reciprocal rule
1
g 0 (x)
d
=−
, if g(x) 6= 0.
dx g(x)
(g(x))2
Question 3. Use the reciprocal rule to find the derivative of h(x) = x−n , where n is a positive
integer.
Question 4. Use the reciprocal rule to prove the quotient rule.
Question 5. Suppose that we know that for every x and h, a function f satisfies
f (x + h) − f (x) = −hx2 − 6hx − 3h2 x + 2h2 + 4h3 .
Find f 0 (x). Do you know of any function f satisfying equation (1) for every x and h?
Question 6*. (Optional, just for fun) Show that
100n n20
√
= 0.
n→∞
n!
lim
1
(1)