Name: ________________________________ Calculus
Derivative Practice
Find the derivatives of the given functions, and simplify.
1.
f ( x ) = sin x
2
4.
y = tan x
7.
f ( x ) = ( sin x )
10.
4
f ( x ) = ⎡ cos ( 5x + 3 ) ⎤
⎣
⎦
5
(
f ( x ) = cos ( 2x )
3
2.
) (t
3
5
)
4
6.
8.
y = sin (13x )
9.
k (x) = 3 1 + x
11.
f (z) =
12.
s (t) =
14.
G ( x ) = ( 3x − 2 )
15.
y = 1 + cos x
17.
y = x +1
x +2
18.
⎛x⎞
y = csc ⎜ ⎟
⎝3⎠
2
21.
y = sin ( cos ( 4x ) )
24.
y = cos sin x
9
1
5
2z − 1
y = ( 2x − 5 ) 8x − 5
19.
y = sec ( 2x ) − tan ( 2x )
20.
y = cot 1 + x
22.
y = sin ( sin ( sin x ) )
23.
y= x+ x+ x
25.
3
3
f ( x ) = ⎡ x + ( 2x − 1 ) ⎤
⎣
⎦
28.
(
4
2
2
)
−3
2
v=x x+
3
26.
10
(
2
2
3
y=
1
x
3
y = 2x sin ( 3x )
16.
−7
2
5.
g ( t ) = 6t + 5
3
f (x) = 9 − x
⎛ 2 1⎞
y = ⎜x − ⎟
x⎠
⎝
13.
2
3.
)
3
(5x
2
− x +1
)
12
2
3
4
t +1
3
t −1
(
2
)
6
2
(
2
)
2
x
27.
sin x + cos x
y=
x tan x
sec x
x
For the following, find the equation of the tangent line to the function at the given point.
2
29.
y = 25 − x
31.
⎛ πx 2
y = tan ⎜⎜
⎝ 4
3
at ( −3, 4 )
⎞
⎟⎟ at (1,1 )
⎠
30.
⎛ x ⎞
y=⎜
⎟ at ( 2, 8 )
⎝ x −1 ⎠
32.
⎛π ⎞
y = sin x + cos 2x at ⎜ ,1 ⎟
⎝6 ⎠
33.
Find all points on the graph of the function f ( x ) = 2 sin x + sin x at which the tangent line is horizontal.
34.
Suppose that F(x) = f(g(x)) and g(3) = 6, g’(3) = 4, f’(3) = 2, and f’(6) = 7. Find F’(3).
35.
The curve y =
2
1
(1 + x )
2
is called a witch of Agnesi. Find an equation of the tangent line to this curve at
1⎞
⎛
the point ⎜ −1, ⎟ .
2⎠
⎝
36.
How many tangent lines to the curve y =
lines touch the curve?
x
x +1
pass through the point (1, 2)? At which points do these tangent
Find the equation of the normal line to the curve at the given point. The normal line to a curve C at a point P is the line
that passes through P and is perpendicular to the tangent line to C at P.
37.
y =1− x
2
at ( 2 , −3 )
38.
y=
1
x −1
at ( 2,1 )