Problem Set 7
Due 2:00PM Sep. 12
1
Additional Differentiation Rules
We did not prove the following propositions yet. We are going to prove on Monday. Feel free to
use the following results for this problem set if needed.
Proposition 1.1 Suppose that f (x) = lnx. Then f 0 (x) = x1 .
Proposition 1.2 For any positive base b,
d x
dx a
= (ln a) (ax ).
Proposition 1.3
d
(sinx) = cosx
dx
d
(cosx) = −sinx
dx
d
(tanx) = sec2 x
dx
2
d
(cscx) = −cscx · cotx
dx
d
(secx) = secx · tanx
dx
d
(cotx) = −csc2 x
dx
Questions
1. Find the derivatives of the following functions.
• f (x) = e5x−4
√
• f (x) = ln 3x2 + 2x
• f (x) = e3x−2
• f (x) =
(x−x2 )
e2x
• f (x) = ln(g(x2 ))
• f (x) = eg(2x)
• f (x) = g(x) + h(y)
• f (x) = (xδ + aδ )γ
2. Find the derivatives of the following functions.
• f (x) = 186.5
• f (x) = 5x − 1
• f (x) = x2 + 3x − 4
1
• f (t) = 14 (t4 + 8)
• f (x) = x−2/5
• V (r) = 34 πr3
• Y (t) = 6t−9
√
• G(x) = x − 2ex
• F (x) = ( 12 x)5
1
x2
• g(x) = x2 +
• y=
x2 +4x+3
√
x
• y = 4π62
• y = ax2 + bx + c
1
√
4 3
t
• v = t2 −
3. Differentiate
• f (x) = x2 ex
• y=
ex
x2
3x−1
2x+1
= (2x3
• g(x) =
• V (x)
• F (y) =
• y=
• y=
1
y2
+ 3)(x4 − 2x)
− y34 (y + 5y 3 )
t2
3t2 −2t+1
(r2 − 2r)er
√
v 3 −2v v
v
y = x4 +x1 2 +1
f (x) = x+x c
x
• y=
•
•
4. Differentiate
• y = sin4x
• y = (1 − x2 )1 0
• y=e
√
x
• F (x) = (x3 + 4x)7
√
• F (x) = 4 1 + 2x + x3
• g(t) =
1
(t4 +1)3
• y = cos(a3 + x3 )
• y = e−mx
• g(x) = (1 + 4x)5 (3 + x − x2 )8
• y = (2x − 5)4 (8x2 − 5)−3
• y = xe−x
2
• y = excosx
2
• F (z) =
• y=
q
z−1
z+1
√ r
r2 +1
• y = tan(cosx)
• y = 2sinπx
• y = (1 + cos2 x)6
• y = sec2 x + tan2 x
• y = cot2 (sinθ)
p
√
• y = x+ x
√
• sin(tan sinx)
3