.
Derivative :Chain Rule (Examples List)
1. Find
dy
if
dx
(a) y = u12 and u = 5x2 3x + 2
p
x+2
(b) y = u and u =
x 2
p
u+2
(c) y =
and u = x
u 2
x2
u
(d) y = e and u =
2
2. Find the following derivatives.
dy
x
if y = sin u and u =
dx
3
dv
(b)
if v = cos s and s = et
dt
dE
1
(c)
if w = and z = q 2 + R2 ; where R is a positive constant.
dq
z
(a)
3. Use the chain rule to …nd the derivative of y = (sin x + cos x)5 . Do it in two ways: (a)
by introducing a “middle”variable u; and (b) by using the “outer-inner”procedure.
4. Find the derivative of the following functions.
f (x) =
p
3
x4
+
x2
+ 1; g (x) =
x
2
x +1
12
q
p
; h (x) = x + x:
5. Di¤erentiate the following functions.
R ( ) = cos (4 ) ; F (t) = sin (sin t) ; G (w) = tan (5ew ) ; y = sec (2x
3) :
6. Find the derivative of the following functions.
y = sin2 x; z = tan5 x; w = sec3 x; u = ekx (where k is a non-zero constant)
7. Di¤erentiate the following functions and simplify your answers.
y = x2 + 3x
2 e x ; y = x2 sin
3
1
x
8. Di¤erentiate y = (2x + 5)4 (x2 + 1) and factor your answer.
x
9. Di¤erentiate f (x) = p
and simplify your answer.
4 x2
10. For f (x) = (x2 + 5x + 4) e3x ; …nd f 0 (0) and f 00 (0) :
11. Find the third derivative of y = tan2 x:
1
12. Find an equation of the tangent line to the graph of y =
p
x3 + 1 at the point (2; 3) :
t
13. Suppose that the position of a moving particle is given by s (t) = e t sin ; where s is
2
in meters and t in seconds. Find its velocity and acceleration at time t = 1:
14. Suppose that air is being pumped into a spherical balloon so that its volume V is
increasing at a constant rate of 36 cm3 per second. How fast is its radius r increasing
4 3
at the moment when its radius reaches 3 cm? [Volume of a sphere: V =
r :]
3
p
15. If g is a di¤erentiable function and f (x) = x2 + 4g (x); how is f 0 (x) related to g (x)
and g 0 (x)? Find f 0 (3) if g (3) = 4 and g 0 (3) = 5:
16. If g is a di¤erentiable function and f (t) = g (sin 3t) ; how is f 0 related to g 0 ? Find f 0 (0)
if g 0 (0) = 4:
17. Suppose that f and g are di¤erentiable functions and some of the values of f; g; f 0 and
g 0 are known to us:
x
1
2
3
4
f (x)
2
4
1
3
g (x)
3
1
4
2
0
f (x) 5 11
3 7
g 0 (x)
4 9
6 10
Find (f
g)0 (2), (g f )0 (4) ;
d
(f (2x))
dx
and
x=1
dp
g (x)
dx
dy
= ky; show that y = e
18. If y = f (x) satis…es the equation
dx
derivative.
Joseph Ling (Fall 2013)
2
:
x=3
kx
f (x) has constant zero