Waldron
Ch Mrs. 3.3
BC Calculus 604
Rules of Differentiation
1. Given M = at 3 - c 2 t +
2.
t
dM
dM
, find a)
and b)
a
dt
dA
x2 + 3
Find an equation for the tangent line to f(x) =
at x = 2, and
2x
also for the line perpendicular to the graph at x = 2.
#3-4: Find f '(x)
3. f (x) = (x 2 - 1)(x + 21)
4. f (x) = (4x 2 + 1)(2x 4 - 2)
5. Find an equation for the line tangent to the curve y =
x2 - 1
at the
3x
point (1,0). Support your answer graphically.
6. The volume V of a sphere of radius r is
4
r 3 . Notice that
3
dV
= A . Explain in terms of geometry why the instantaneous rate of
dr
change of the volume with respect to the radius should equal the
surface area.
7. Find the derivative eighth of f (x) = x 7 + 5x 5 - 4x 3 + 6x - 7 .
Think ahead! (the nth derivative, f(n)(x), is the result of differentiating
f(x) n times).
Find the seventh derivative of f(x).
8. Find f '(x) if f (x) =
x2 - 1
x2 + 1
9. Find f '(x) if f (x) =
2x
x+3
a
2
-R
dC
dC
b
10. Find
and
if C =
dR
dT
nT
#11-12 Find f '(x)
11. f (x) = (1 - x) (1 + x 2 )
12. f (x) =
(x + 1) (x + 2)
(x - 1) (x - 2)
1
13. Suppose that u and v are differentiable functions of x at x = 2 and
u(2) = 3, u '(2) = -4, v(2) = 1 and v '(2) = 2. Find the following
derivatives:
a)
d v
dx u
b)
d
( 3u - 2v + 2uv )
dx
14. Find the tangent to the Witch of Agnesi, y =
(2,1).
8
at the point
4 + x2