Math 2144 Gateway Quiz Information
Spring 2015
INSTRUCTIONS: The gateway quiz on differentiation is worth 5% of your grade. It is
a closed book quiz with no calculator usage and no other notes or aids allowed. No
partial credit may be earned. Students completing six of the seven problems correctly
will earn full credit; all others will score 0 points.
The differentiation gateway is a 30 minute quiz scheduled for Monday, March 2nd
during class. There will be additional opportunities to retake the gateway throughout March
for those who do not pass on the first attempt, but these will take place outside of class
time. No gateways will be given after March 31st. Your differentiation skills must be
accurate and fast to facilitate your progress in the rest of the course.
Topics covered by the Differentiation Gateway
(from sections 3.2 – 3.9 of the text)
1. For any constant c,
d
(c) = 0
dx
2. The Power Rule: For all exponents n:
d n
x = nxn−1
dx
3. Linearity Rules: Assume that f and g are differentiable.
Sum and Difference Rules: f + g and f − g are differentiable, and
(f + g)0 = f 0 + g 0
(f − g)0 = f 0 − g 0
Constant Multiple Rule: For any constant c, cf is differentiable and
(cf )0 = c(f 0 )
4. The Product Rule:
d
(f (x)g(x)) = f (x)g 0 (x) + f 0 (x)g(x)
dx
5. The Quotient Rule: given g(x) 6= 0, then
d
dx
f (x)
g(x)
!
=
g(x)f 0 (x) − f (x)g 0 (x)
(g(x))2
6. The Chain Rule:
d
(f (g(x))) = f 0 (g(x)) g 0 (x)
dx
The Chain rule in Leibniz notation: let y = f (g(x)) and u = g(x) then
dy
dy du
=
·
dx
du dx
1
Math 2144 Gateway Quiz Information
Spring 2015
Table 1: Some Important Derivative Formulas to Know
d n
d
1
(x ) = n xn−1
(ln x) =
dx
dx
x
d x
d x
e = ex
b = (ln b) bx
dx
dx
d
d
sin(x) = cos(x)
cos(x) = − sin(x)
dx
dx
d
d
tan(x) = sec2 (x)
cot(x) = − csc2 (x)
dx
dx
d
d
sec(x) = sec(x) tan(x)
csc(x) = − csc(x) cot(x)
dx
dx
d
1
1
d
arcsin(x) = √
arctan x =
2
dx
dx
1 + x2
1−x
Sample Gateway Quiz on Derivatives
Solutions need not be simplified.
1. Find
dL
if L(p) = (−p4 − 2p + π)5 .
dp
2. Find y 0 if y =
arcsin(x + 1)
.
ln(x)
3. Find
dW
if W = e−4 − e4t .
dt
4. Find
dy
if y = (3x + 8) cos(8x + 3).
dx
5. Find B 0 (x) if B(x) = 10(tan(x))−5/4 .
6. Find
dy
ln(x + 3)
if y =
.
dx
x2
7. Find f 0 (x) if f (x) = A ln(sin(Bx)), where A and B are constants.
2
Math 2144 Gateway Quiz Information
Spring 2015
Solutions to Practice Gateway Quiz
dL
if L(p) = (−p4 − 2p + π)5 .
dp
Solution: Using the chain rule,
1. Find
dL
= 5(−p4 − 2p + π)4 (−p4 − 2p + π)0 = 5(−p4 − 2p + π)4 (−4p3 − 2).
dp
arcsin(x + 1)
.
ln(x)
Solution: Using the quotient rule and the chain rule,
2. Find y 0 if y =
(ln x)(arcsin(x + 1))0 − arcsin(x + 1)(ln x)0
y =
=
(ln x)2
(ln x)
0
√ 1 2
1−(x+1)
(1) − arcsin(x + 1) x1
(ln x)2
dW
if W = e−4 − e4t .
dt
Solution: Notice that e−4 is a constant, so, using the chain rule,
3. Find
dW
= 0 − (e4t )(4) = −4e4t .
dt
dy
if y = (3x + 8) cos(8x + 3).
dx
Solution: Using the product rule and the chain rule,
4. Find
dy
= (3x+8)(− sin(8x+3)(8))+cos(8x+3)(3) = −8(3x+8) sin(8x+3)+3 cos(8x+3).
dx
5. Find B 0 (x) if B(x) = 10(tan(x))−5/4 .
Solution: Using the chain rule,
B 0 (x) = 10
−
5
25
(tan x)−9/4 (sec2 x) = −
(tan x)−9/4 sec2 x.
4
2
dy
ln(x + 3)
if y =
.
dx
x2
Solution: Using the quotient rule and chain rule,
6. Find
x2
dy
=
dx
1
x+3
− ln(x + 3)(2x)
(x2 )2
=
x2
x+3
− 2x ln(x + 3)
x4
7. Find f 0 (x) if f (x) = A ln(sin(Bx)), where A and B are constants.
Solution: Using the chain rule twice, we compute that
!
1
AB cos(Bx)
f (x) = A
(cos(Bx))B =
.
sin(Bx)
sin(Bx)
0
3
.