Derivative Exercise:
dy
, as appropriate:
1. Find f 0 (x), or
dx
a. f (x) = 3
b. f (x) = 4x + 1
c. f (x) = −2x2 − 3x + 1
3x3
4
√
e. f (x) = x − 1
√
√
√
f. f (x) = − x5 + 3 x3 + 7 x
d. f (x) =
g. y = sin x
h. f (x) = sin
3π
4
i. f (x) = ex
j. f (x) = e
k. f (x) = 3xex
l. f (x) = 3x sin x + sin x cos x
m. f (x) = 3x2 ex + 4x tan x
n. f (x) = sin x cos xex
o. y = 3x2 sin x cos xex
3ex − sin x
p. y =
cos x − 4ex
√
x+1
q. y =
3x − 4
3ex − sin x
r. y =
cos x − 4ex
9x2 ex + 4x cos x
√
s. y =
2 sin x + xex
t. y =
ex sin x − x cos x
3x cos x + 4xex
u. y = (4x + 3)3
v. y = (x2 − 3x − 2)5
√
w. f (x) = 4x3 − 2x2 + 4x + 5
x. f (x) = sin(4x)
y. f (x) = e3x−1
z. f (x) = sin(3x2 + 1)
aa. f (x) = cos(sin(4x − 2))
bb. y = sin(e3x )
cc. y = ex
2
dd. y = sin3 x
ee. y = cos4 (x2 + 1)
2
ff. y = ecos
x−1
gg. f (x) = tan3 e2x−1
hh. f (x) = cos(2x3 − 4)ecos x
ii. f (x) = x tan(x2 + 1)e3x
2
jj. f (x) = x2 esin x + ex sin(cos(3x − 1))
2
e + 3xex −2
kk. y =
e − 4 sin x cos2 x
2
2x cos(x2 − 4x) + e3x −1
ll. y = 2
3x − 4e3x cos2 (4x+1) + sin3 (5x + 1)
mm. y = arccos x
oo. y = ln x
pp. y = ln(x2 + 4x − 1)
qq. y = arcsin(x2 − 5x + 1)
1
x
ss. y = ln(ln x)
rr. y = ln
1
ln x
uu. y = arctan2 (ln 3x)
tt. y =
vv. y =
ln(3x)
sin(ln x2 )
ww. y =
3e4x sin−1 (3x2 + 1)
ln x sin 5x + 3x2 cos2 x
2. Find the equation of the line tangent to the given function f at the given point
x = a:
a. f (x) = 2, a = 3
b. f (x) = 5x − 1, a = −2
c. f (x) = x2 − 2x + 4, a = 1
π
d. f (x) = sin x, a =
6
e. f (x) = ln x, a = 3
f. f (x) = e3x , a = −2
g. f (x) = xex − ln 4x, a = 1
3. a. Define a function that has a left and right handed limit at a point a, but f
does not have a limit at a.
b. Define a function that has a limit at a point b but is undefined at b.
c. Define a function that has a limit at point c, is defined at c, but is discontinuous
at c.
d. Define a function that is continuous at point d but is not differentiable at d.
e. Define a function that is first differentiable at point p but is not second differentiable at point p.
f. Draw the graph of a function which has all of the properties in a, b, c, d, e, above.
4. Think About:
a. Given the graph of the original function f (x), how can you draw the graph of its
derivative, f 0 (x)?
b. Given the graph of the derivative of a function, f 0 (x), how can you draw the
graph of the original function, f (x)?
dy
by implicite differentiation:
dx
a. y sin x = x cos y
5. Find
b. x2 y + y 2 x = 5
c. 2y − 4x sin x = ey
x
d. ln(xy) + = 1
y
e. exy + ln(x + y) = 2xy
6. Find f 00 (x) and f 000 (x):
a. f (x) = 4
b. f (x) = 3x2
c. f (x) = ex
d. f (x) = sin x
e. f (x) = ln x
f. f (x) = arccos x
g. f (x) =
x2 + 1
x−1
h. f (x) = esin x
√
i. f (x) = x
j. f (x) = x5/4
7. Find the linearization of the given function at the given point a:
a. f (x) = 2x − 3, a = 2
b. f (x) = 2x − 3, a = 4
c. f (x) = 2x − 3, a = 10
d. f (x) = 4x2 − x + 3, a = −1
e. f (x) = sin x, a = 0
π
f. f (x) = cos x, a =
4
x
g. f (x) = e , a = 0
h. f (x) = ex , a = 1
i. f (x) = ln x, a = 1
j. f (x) = ln x, a = 2
k. f (x) = arctan x, a = 0