Section 2.3 The Chain Rule
#1-10: Find f[g(x)], and don’t simplify your
answer!!!
1) f(x) = x3; g(x) = x2 + 2x + 1
2) f(x) = x4; g(x) = 4x2 + x - 5
3) f(x) = √𝑥;
3
g(x) = 2x – 7
4) f(x) = √𝑥 ; g(x) = 3x – 1
5) 𝑓 (𝑥 ) = 𝑥 2⁄3 ; g(x) = x2 +4
6) 𝑓 (𝑥 ) = 𝑥 4⁄5 ; g(x) = x2 + 4
7) f(x) = ex; g(x) = x2 + 2x + 1
8) f(x) = ex; g(x) = 4x2 + x - 5
9) f(x) = ln(x); g(x) = 3x + 5
10) f(x) = ln(x); g(x) = 2x-7
#11-20: Create two functions f(x) and g(x) such that
h(x) = f[g(x)]
11) h(x) = (x-3)2
12) h(x) = (2x-5)3
13) h(x) = (𝑥 − 4)
14) h(x) = (𝑥 −
1⁄
2
1⁄
2) 3
15) ℎ(𝑥) = √𝑥 + 5
3
16) ℎ(𝑥) = √7𝑥 + 1
17) h(x) = (x-3)2 + 4(x-3) + 1
18) h(x) = 4(x-1)2 + 6(x-1) + 4
19) h(x) = 2(x4 + x + 1)3 + 3(x4 + x + 1)2 – 3
20) h(x) = 11(x3 + 3x)3 + 3(x3 + 3x)2 – 2
#21-32: Use the Chain rule to find the derivative of
each function.
If h(x) = f[g(x)] then h’(x) = g’(x)*f’[g(x)]
21) h(x) = (2x-4)3
22) h(x) = (5x – 3)2
23) h(x) = 5(7x + 1)4
24) h(x) = 3(8x + 7)5
25) h(x) = 3(2x – 1)-4
26) h(x) = 2(5x – 6)-3
27) ℎ(𝑥) = (𝑥 2 + 6𝑥 + 1)3
28) ℎ(𝑥) = (3𝑥 2 − 5𝑥 + 2)3
29) ℎ(𝑥 ) = √3𝑥 − 5
30) ℎ(𝑥 ) = √𝑥 2 + 3
3
31) ℎ(𝑥 ) = 4 √5𝑥 + 1
3
32) ℎ(𝑥 ) = 7 √4𝑥 + 1
#33-46: Find the derivative of each function.
33) 𝑦 = 5𝑥 (2𝑥 − 4)3
34) 𝑦 = 5𝑥 2 (7𝑥 + 1)3
35) 𝑔(𝑡) = (𝑡 2 + 6𝑡 − 1)(2𝑡 + 5)2
36) 𝑔(𝑡) = (3𝑡 2 + 5𝑡 − 1)(4𝑡 − 1)2
37) h(y) = (3y + 1)2(6y – 3)
38) f(y) = (2y – 3)(4y + 1)2
39) 𝑦 = 3𝑥 2 √2𝑥 − 5
3
40) 𝑦 = 4𝑥 √7𝑥 + 1
41) 𝑦 =
42) 𝑦 =
2
(3𝑥−4)4
5
(2𝑥−9)3
43) 𝑓(𝑥 ) =
44) 𝑓(𝑥 ) =
(3𝑥−1)2
2𝑥−5
(4𝑥+5)2
2𝑥+1
45) 𝑔(𝑥 ) =
46) 𝑔(𝑥 ) =
𝑥2
√2𝑥−3
𝑥3
√5𝑥−7
#47-52:
a) Find all values of x where the tangent line is
horizontal
b) Find the equation of the tangent line to the
graph of the function for the values of x found in
part a.
47) f(x) = (2x-3)2
48) f(x) = (3x – 4)2
49) y =4x (x – 1)2
50) y =3x (x – 2)2
51) y = 5(x + 3)4
52) y = 7(5x – 6)2